In order to use this guide to quickly level your Skyrim skills you will need understand a few basic game mechanics first. Killing things and completing quests does not give you experience like a typical RPG would. Instead, gaining experience is the result of leveling up your skills. The amount of experience you need to level up increases the higher in level you are, meaning at level 1 let us pretend you need 100 XP to get to level 2. You might then need 120 XP to get to level 3, etc. Also, the higher your skill is, the more XP you get when you level it up, meaning that raising Archery from level 1 to 2 might grant you 50 XP, but leveling Archery from level 2 to 3 might give you 100 XP.
Now, the typical way you increase your skill level in something is by using the related skill. For example, to increase your Archery level you need to hit enemies with arrows, to level Sneak skill you need to successfully sneak by an NPC without them detecting you, to level Lockpicking you need to work on picking locks, etc. There is an alternative though.