Always have your 3 mana Gems at the beginning, use them properly.
Start the encounter at melee (in HC warn you RL you are with MDPS), do your opener as usual and on the 1st wave of ghouls use your sapper charge. When you got it move out of the melee under your GCD and as fast as possible. Go in a location where nobody is and not on the way of the raid, of course at a proper range to be able to correctly join the shambling if you need to be dispelled.
At phase 1: If you are well geared: for the most effective raid dps, only use living bombs on LK, and if you have a very long move to do when you have to go on your transition spot, you can put one on the shambling meanwhile you are moving. If you are low geared do not hesitate to maintain a LB on one of the shambling in order to proc more HS and increase your dps on LK. Use correctly your GCDs to move at the transition !
At the 1st transition: find a spot at the border of the platform between 40 an 41 yards, in order to be able to dps both LK and the Raging Spirits. When a Raging Spirit spawns, put and LB on it and continue your dps on LK. If the raid has issues to shut down Raging Spirits before the next one spawns, alternate between the LK and Spirits.
When the 1st transition ends: do not forget that you have a huge amount of threat on the boss so warn your main tanks before the the 2nd phase or ask your paladins for HOS, it will be enough because approximately 7 sec after the transition end (depending on your raid dps) you will have your images ready, and under images the LK will not target you. When the ground breaks, if you are at 40-41 meter during the transition you will not fall down, so be sure to wait for GCD in order to move in the center of the platform and remain there. You have 10 seconds to be on the pack before Val'kyrs so use these 10 seconds to move and your GCDs in order to always be casting.
To reach your mates at this time, move on GCD (blinking costs you more in term of DPS loss).
At Phase 2 : stay packed with your mates, this is the only time in ICC when you will not apply the 35+ meters rule and loose some HS procs.
When val'kyrs spawn: LB each of them and use the 3 sec given by your gcds to move away from the pack, in order to do not fuck up a defile. After that, single target a val'kyr and remember you have one of the longest range in the raid, so it's your job to shut down the val'kyr which is not well packed ! Use your Saronite Bombs on the Valkyrs, but never use your Sapper Charges (it causes a very small stun and that diminishes the following stuns). Never refresh your living bombs on val'kyrs, it isn't useful for the raid damage, but be sure to continue refreshing Living bomb on LK, even during the time you hit the val'kyrs.
For managing Defile: be sure you have the LK on Focus with Target of Target activate, when the timer tells you Defile will come soon, focus your mind on the current target of the LK and stay in the pack. If the LK targets you for defile instantly blink in the direction of the throne (the throne zone is the better to place a Defile because no valkyrs will never run this way !
Phase 3: depending on the location of the LK, try to maintain the 35+ range from the boss (most of the time being at the exact center is the best way to always have the range to reach LK), place the nicest defiles and respect the 35+ rule both in the same time. But take care about the Vile Spirits and remember if one will reach you, you can Frost nova him or blink away.
Phase 1: after my opener, I'm used to position myself between the MDPS and the RDPS packs, in order to be independent in my movements.
Always stay away from your mates in order to avoid moving at an inconvenient time for a shadow trap.
If a Shadow trap is triggered you instantly blink. That way you will break your flight and won't go out of the platform.
Phase 2: Never put LB on mid-life val'kyr which are in the air, it's a huge dps loss on the boss. Moreover it will make them focus you with their attacks, and even a couple of them can be deadly.
If you managing your mana well with your mana gems, you can reach the FM phase without being Oom, in this case evocate inside Frostmourne at a time it's not dangerous.
For the Frostmourne: prepare macros to target Wicked spirits and then use a living bomb on one, switch, fire blast another and then use Ice Javelin, if you proc HS use a Pyro.
When you are going out of FM blink away to be sure you do not put a bad defile in case you are targeted for it.
Please take in notes: Always single target on LK ,except in cases I mentioned, like on Raging Spirits and on Val'kyr spawns, because in my opinion dps-ing ghouls more than with engineer bombs/charges which are free gcd spells, or whith LB explosions, or even putting LB on a Shambling or refreshing LB on a Val'kyr who will go under 50% before explosion, is a way to gain dps on recount but you will lose effective dps for the raid. Clearing ghouls is the UHdk, Warrior, and Paladin's job, not yours. They will do it perfectly and not by losing any effective dps.
I don't play LK enough as Arcane mage to properly advise you. However know that LK is known as a boss very suitable to the high mobility of the fire mage, and a real curse for the static Arcane mage.