How can I increase weapon damage as much as possible in Skyrim?

Via Exploit:

1. Get Yellow Mountain Flowers
2. Get Salt Piles
3. Buy Soul Gems
4. Create a Fortify Alchemy Item(s)
5. Equip the Fortify Alchemy Item(s)
6. Create & Drink a Fortify Restoration Potion
7. Remove the Fortify Alchemy Item(s)
8. Repeat Step 5 through 8 until the Fortify Alchemy Item has a ludicrous boost
9. Create & Drink a Potion of Fortify Enchanting
10. Enchant any item that has a damage component (ex: Absorb Health)
11. Have fun with your 6-figure+ damage output

Notes:

• You can do the same for Smithing Weapons/Armour and then use this exploit (but it's just so much damage that you don't even see the difference between having a glitched sword + glitched enchantment than just a glitch enchantment on a sword

• If you need to buy stuff from stores you can use the quicksave->kill->reload to refresh store inventory.

• You don't need a high level of enchanting nor alchemy, although one does make the process faster

• Don't go too overboard on the boost to the Fortify Items, as if they become too high the game may crash (~390000% is fine)

• Recipe of Potion of Fortify Enchanting

Source (Youtube)

There are a couple of ways to increase weapon damage.

1. Get a better weapon, unless you are in end game and have some of the best weapons and materials available to, the next few options might not be worth it

2. Upgrade your weapon on the grindstone at a forge, your going to need some materials to do this, and the higher level you are in smithing, the greater the increase in damage.(Upgrade Ranks: Fine, Superior, Exquisite, Flawless, Epic, Legendary)

3. Put damage enchantments on it, this can increase the damage done by the weapon instantly by 10 damage depending on skill and spell, at level 100 enchanting you can put 2 damage enchantments on it, and with a grand soul gem, get even more damage out of it

4. Put poison potions on your weapons, these don't last long, but it will still increase your damage

5. Like one handed weapons, two handed, archery? Go to the specific skill tree of your favorite and put a few skill points there. The first point in each tree is a 20% increase in damage.

P.S. To get absolute MAX DAMAGE you're going to need enchanted equipment with fortify smithing on it and a potion of fortify smithing, as well as smithing at level 100 so you can max the damage increase on the grindstone

If you want more information on Smithing in Skyrim visit: http://elderscrolls.wikia.com/wiki/Smithing_(Skyrim)

NOTE: In the early days of Skyrim there were a couple of damage exploits, but they were promptly patched, to my knowledge there aren't any damage exploits in the game today

Use a potion of smithy and have the respective perk for that material (under the Smithing Perk Tree), then upgrade it at a grindstone. Some alter blessings and items boost smithing. The higher your smithy, the better items will upgrade. No glitches required.