Disadvantage
Disadvantage is often referred to as a hellscape for King K. Rool, and it absolutely can be in many instances. However, with updates, it has improved drastically, and K. Rool can be seen as a competent heavy in this regard. This guide will assist players looking to get out more easily.
Mindset
When players are at a disadvantage, I consistently see them get extremely tilted (a Poker term for anger and frustration) and defeat themselves before they even give themselves a chance to fight back, to the point of me needing to write this section in the first place. This is extremely bad practice, and managing tilt is important; not only for the sake of not making everyone else's day worse but also so you can actually enjoy the game and get those wins going. Take it from an ex-Poker addict: tilt is a toxic, self-destructive habit that is severely detrimental to your gameplay, no matter how unassuming it seems. If you feel tilted, it warrants attention as soon as possible.
Here are some tips to avoid or minimize the results of tilt. Read them carefully, and apply them in your next games;
Play with the game's outcome at the back of your mind - Play to learn, not to win. While this appears counterintuitive, you should ideally be playing with the aim of being the best, rather than focusing on small victories. A single loss does little in that path. If anything, a loss is an improvement opportunity. Interact with your opponent after the game, exchange tips and tricks. Fighting game tournaments are social events as much as they are competitive ones.
Play more conservatively - It's a good idea to go for fewer "big boy" plays if you're not even sure they'll work. Those hard reads you're doing have got to have a purpose, that is, calling out your opponent. Look at what your opponent is doing, respond in turn without committing too hard. This way, you're more likely to avoid advantage.
Consider variance - Video games, especially ones of this genre, have a lot of variance involved. Even the most skilled player will experience an upset at some point, that's just how games work. Both you and your opponent will have a variety of cards you haven't shown yet. These are there to add variance to the game, and you will get surprised by them. It is up to you to predict what these cards are, and if you fail, well, it's gonna be like that sometimes. You've gotta shrug it off, champ. Sod's law.
Check your ego - Players that experience tilt often have a bit of an ego. This isn't to say they all do, I've seen some with inferiority complexes as well. Nevertheless, it is among the most common causes of tilt: not playing to your own ridiculous expectations. I'll put this lightly: your results and wins don't mean jack shit in bracket. All that tells me is you're either a strong player or a weak one. A strong player is easier to scout prior to the match, a weak player is more vulnerable to cheese. That's it. Don't worry about what your past successes are, the only thing that matters is your opponent.
After the game, take a break - Leave the setup, get something to eat and drink, get some fresh air. Take some time to refresh your mindset, then maybe play some CPUs or something on a lone setup and get yourself back into gear.
While few of these particularly apply to disadvantage, all of these are successful ways to develop a healthy mindset that allows you to play better overall. You've probably heard at least one before, and these don't consider any underlying health conditions, but they are ways top players have succeeded in bracket.
Landing
This video is great, especially since it's applied to a fellow big body in ROB, and I don't really have anything to add. King K. Rool has one of the best times landing relative to the other superheavies, simply because NAir is so ballbustingly good. You can often get by in low to mid-level play simply by mashing NAir, but as you rise up, players will begin to catch on. Let's review your landing options.
NAir
Since it was already cited, let's review this first. Armor on frames 6-30, 9 frames of landing lag, ridiculous combo potential upon landing...yep, this is certainly a doozy. It's not uncommon to use NAir to land, armor through a multihit, and then get a Late NAir -> Grab or something stupid. This move is absolutely incredible and is one of the best ways to leave disadvantage.
Remember that you can buffer dash through NAir's landing lag, or do well, anything. Players often just let themselves get grabbed when landing when they could instead be spotdodging or dashing away with relative impunity. You need to think ahead, just because you've landed doesn't mean you're out yet.
So with this in mind, NAir is your premier reversal tool. If you're looking to leave disadvantage and go straight into advantage, well, this is it. However, I would argue that because of its nature as a reversal, this means it's one of those cards you should actually keep hidden. NAir is best-used when you're looking to reverse the situation in a way that would give you a better advantage state, such as if it would launch the opponent off-stage or put them in a situation where you can combo them. This is your best tool, but best hardly means "spam".
Gut Check
People love to hate on this move, and as established when going over moveset, Gut Check isn't bad. In fact, it's one of the better combo breakers in your kit, since frame 4 intangibility is stupid. If it works, your opponent is taking a whole 1.5x damage back with amazing knockback and a hefty frame minus. If you watch where your opponent's combo is going, you can often predict when an attack is hitting that counterbox and belly bounce them back to Smash 4. Try it on a Quickplay Ness attempting DThrow -> FAir at 0%, it'll be just gravy.
Fast Fall Neutral Airdodge
Let's call this FFNA for this section. This is a technique you must learn when playing a fast faller. Jump from a high platform, start a fast fall, return your stick to neutral and do an airdodge. See that fall speed while being intangible? Yeah? Now do that in disadvantage. It'll require you burn your double jump, but for K. Rool, this is inconsequential. This fat bastard falls like a brick and you'll be able to skip past so many juggle attempts with this. You're up there with King Dedede in how valuable this is to disadvantage. Remember to airdodge to ledge sometimes too!
These are your main ones, but there are more niche things you can do; FAir, DAir, etc. However, they shouldn't really be used as a standard and moreso in niche situations where they're appropriate.
Escaping Combos
The first step to escaping combos is to not shut off in disadvantage, then watch your opponent and where you're going. Refer to the mindset section for assistance with that. You need to identify the combo the opponent is going for, as well as whether you're headed for a juggle situation or an actual combo.
Your primary combo escape tools as Gut Check, airdodge, and NAir.
Gut Check and airdodge should be alternated based on where the combo is going. The reward for landing Gut Check is massive: it resets neutral or sends the opponent offstage and dishes back 1.5* damage, which is flat out insane. If the combo is headed for K. Rool's belly, that move is yummy, give em what-for. On the flip side, airdodge can be used otherwise to score a tech if you land, which can be helpful for just getting to breathe for a bit. Otherwise, it's still fine.
NAir is considered to be reliable, but has frame 6 armor, so oftentimes a combo won't leave an opening for it unless the opponent is overextending. Ergo, this is moreso a landing tool than a combo escape option, and should be treated as such. Generally, I don't recommend spamming this as a combo escape tool, it's really weird.
Don't jump in these situations, as oftentimes you'll either lose it, or if you get it you'll put yourself in a juggle situation. This should be an absolute last resort.
Getting off the ledge
To preface, K. Rool is an absolute champ at ledgecamping and you should learn to be patient on the ledge because of this. Many characters are forced to commit when taking this fat bastard off the ledge, allowing you to react in turn with a ledge option. However, K. Rool is generally quite weak at actually getting off the ledge, but don't switch off and go "oh it's impossible then" because he's still got options and can get away with murder.
The central aim of getting off the ledge is reclaiming centre-stage. The second you're back there, you're back to neutral. Don't let the opponent distract you from that goal. Doing something as little as jumping behind them is enough to turn the game back in your favour. Remember to mix up neutral getup, roll, jump, and ledge release -> UAir. Here are some of my favourite options for getting off the ledge, but always experiment.
Ledgedrop -> UAir / Rising FAir
Rising FAir is by far my favourite tool for getting off the ledge. While it's hard to time, with practice, this becomes your best friend. This is because when used in this context, for some magical reason, gives you maximum hit advantage on hit and shield. Thus, you get your -5 on shield, +15 on hit at 0 FAir if the opponent lets this touch them. This, for reference, can let you roll behind or even Jab most OOS options, or spotdodge a grab. While you shouldn't spam this - especially if it stales and loses some shield advantage in the process - it's by far one of your best tools. Just be careful if it gets challenged.
UAir is often used by 808 when the opponent tries to catch out his ledgecamping, and should be more widely considered. It lets you get onto the ledge with minimal lag while also being a kill button. If the opponent even tries to contest your ledgecamping, reacting with this is pretty much free neutral unless they shield it. Even then, you can drift behind smaller characters after hitting shield or rapidly move away, making it decently safe to boot unless they specifically know how to punish it.
Ledgejump -> Buffered NAir
NAir is a very nice buff to Rool's ledgejump, being armored and having nice reward on hit. If they're right by the ledge, they'll also get hit, which is incredible for you, as the hitstun is ridiculous and more than enough to let you get back to neutral. You can also followup with a second NAir if you want for added security, especially if fast falling. Remember to buffer dash once you land if doing that!
Neutral Getup -> Jab / FTilt / Dash Attack / BThrow
Jab out of neutral getup is your best offensive option coming off the ledge: it's your fastest grounded option while being just about big enough to catch most options. Recall that this also lets you mix up shields by nature, and if they hold it you can sometimes sneak a roll behind and take back stage control. If that doesn't do it or you need some armor, FTilt or DA will work just fine.
If the opponent is at a high percentage, they need to respect the threat of BThrow. There is always a chance of you randomly taking their stock because they decided to do a shield-based ledgetrap.
Not getting edgeguarded
This will be a bit brief as it's really not that hard. Watch your opponent. If they're going for an edgeguard, unless they're stupid (they probably are) they are aiming to call out your recovery routes and end your stock early. You can tell if they're stupid by if they are just running off with an aerial rather than actually looking at where you are.
Here are some rules to follow for recovering. Let's call them the Five Kommandments, because you're going to follow these religiously without question. Take it as gospel.
Never recover diagonally with Propellerpack unless you absolutely have to. This route is shit and leaves you vulnerable from the front, and doesn't really do anything for you. Oftentimes you'll even miss the ledge and SD when you could otherwise recover. This is because swerving with Propellerpack causes you to lose vertical distance.
Don't autopilot jump, ever. K. Rool doesn't need to jump when recovering unless you're in a really shitty spot. UAir serves as your jump, wait around half a second after being launched and you'll get the hop.
Use Gut Check to stall, especially after being launched. It halts your momentum and allows you to buffer UAir, while also allowing you to watch where the opponent is going. You can also do this to preserve airdodge.
HUG THE STAGE! If you can go to hug the stage, do it. This prevents you from being attacked from the front - where you're most vulnerable - while allowing you to swerve back to contest stage spikes. It is very, very difficult to edgeguard K. Rool with many characters if he does this.
Hold down until the last second when using Propellerpack, as this allows you to use LSI to reduce launch speed upon being hit. This is incredible, as it means stage spikes are nigh-impossible on K. Rool so long as you tech well. For scale, LSI allows you to survive Lucina's BAir until 157%. Because of this near-invincibility against stage spikes, any opponent stage spiking an educated K. Rool is simply giving up stage control and allowing you to return to neutral. You should be welcoming these attempts with open arms.