Dragonsreach in the distance
If, for some reason, you want to walk up to people before you kill them, try some of this.
Spellbreaker & Windshear
The damage is respectable and the defense is great (though you don't really need it.)
Get all the relevant perks in the One-Handed skill.
Get all the perks in the Block skill.
Enchant your necklace, gauntlets, ring, and boots with Fortify One-Handed.
Optionally, enchant your weapon with Absorb Stamina for unlimited power attacks.
Your Light Armor skill perks don't care about the type of shield you're using.
A Daedric Sword provides the best damage output. Use whichever shield you like.
Alternatively, consider these unique items:
This shield--a daedric artifact--offers great magic resistance.
Talk to Kesh the Clean at the Shrine to Peryite, halfway between Markarth and Dragon Bridge. (If you meet an Afflicted Regugee, they'll point you there.) Complete The Only Cure quest to be awarded Spellbreaker.
Improving Spellbreaker at a Workbench does not benefit from any Smithing skill perks.
This scimitar staggers on every hit, making most battles laughably easy.
Ask any barkeep about rumors to hear about Aventus Aretino in Windhelm. Talk to Aventus to start the Dark Brotherhood quest line, and follow that nearly to the end, onto the ship The Katariah. Get up on deck and yank Windshear out of the bowsprit.
Improve it at a Grindstone with the Steel Smithing skill perk.
Dragonbone Sword & Dagger
For the highest damage per second, this is the way.
Pressing both attack buttons simultaneously will execute a dual attack combo, the damage of which is mostly determined by the weapon in your right hand, the speed mostly by the weight of the weapon in your left hand. Hold the buttons down for a dual power attack combo.
Get all the relevant perks in the One-Handed skill.
Enchant your necklace, gauntlets, ring, and boots with Fortify One-Handed.
Enchant at least one of your weapons with Absorb Stamina for unlimited power attacks.
Optionally, keep the dagger unenchanted and use the Elemental Fury shout. You can still enchant the sword.
You can't block, so you may need to dance around a little to avoid blows.
The weapons with the best mix of speed and power are:
Dragonbone Sword in your right hand.
Dragonbone Dagger in your left hand. (Mehrunes' Razor also has its advantages.)
There is no easy way to re-equip an offhand dagger after switching to a bow or spell. You'll need to select it manually in your Favorites menu or Inventory.
Dragonbone Battleaxe
Two-handed weapons hit slightly harder and are more likely to stagger opponents. However, their damage output suffers from their lack of speed.
Get all the relevant perks in the Two-Handed skill.
Enchant your necklace, gauntlets, ring, and boots with Fortify Two-Handed.
Either enchant your weapon with Absorb Stamina for unlimited power attacks, or don't enchant it and use the Elemental Fury shout and Vegetable Soup.
Which weapon is best is debatable. Popular options are:
Forge it yourself.
This unique warhammer is the second-fastest two-handed weapon in Skyrim.
Go to Liar's Retreat, a dangerous dungeon halfway between Markarth and Solitude. The hammer is on the altar.
Improve it at a Grindstone with the Orcish Smithing skill perk.
The daedric artifact of Malacath. It swings slightly faster than a regular warhammer, comes with a hefty Absorb Stamina enchantment, and looks particularly badass.
Go to the orc camp Largashbur on the southern border of Skyrim, southwest of Riften. Help them with their giant problem, then talk to Atub, their wisewoman, to start the Cursed Tribe quest.
Improving Volendrung at a Grindstone does not benefit from any Smithing skill perks (unless you have the Unofficial Skyrim Patch, in which case it's Dwarven Smithing.)
Daedric Gauntlets
Unarmed combat has a variety of funny finisher moves. The damage output is good at the start but drops off later on.
Be Khajiit for a massive claw damage bonus.
Use Heavy Armor for the Fists of Steel perk. (Improving your gauntlets at a Workbench does not improve damage.)
Get the One-Handed skill perk Dual Flurry. (The Armsman perk does not help.)
Go into the Ratway in Riften, grab the Gloves of the Pugilist from Gian the Fist, and disenchant those to learn the Fortify Unarmed enchantment. Put that on your gauntlets and ring.