The simplest way to become a Werewolf in Skyrim is to complete part of the Companion questline which eventually has you become a Werewolf.
Another method is to get the Daedric artifact
The Ring of Hirscine which is part of the
Ill Met by Moonlight quest. Details how to start the quest can be found in the below trophy guide.
NOTE: The benefits of the Ring of Hirscine is you can become a werewolf as many times as you want, as opposed to the normal once per day.
Essentially what I ended up doing is running around Skyrim (aimlessly because you can't access your map) and killing everything. First thing you want to level up is the right hand side of the skill tree. You will eventually unlock the Savage Feeding perk. This will allow you to feed off wolves, bears, trolls, and most other creatures.
You
can feed off Falmer.
The only things you can't feed off are Draugr, Skeletons, Vampires, Ghosts, Daedra, Spriggans, Dragons and other similar creatures.
Once you have the Savage Feeding perk, leveling up is much easier. Random packs of bandits, Foresworn, wild animals and creatures, as well as StormCloak camps will get those perks up fast.
This is how many feedings you need to earn the perks.
NOTE: You do not need to be the one to kill the person/creature to feed off them. I have entered caverns and caves with random bandit bodies laying around and was able to feed off them.
5 feeding - 1st perk
7 more feedings - 2nd perk
8 more feedings - 3rd perk
9 more feedings - 4th perk
12 more feedings - 5th perk
14 more feedings - 6th perk
16 more feedings - 7th perk
18 more feedings - 8th perk
20 more feedings - 9th perk
22 more feedings - 10th perk
24 more feedings - 11th perk
You can apparently cast Soul Trap on the bodies of creatures/people you have already fed on, and the game will register it as an additional feeding.
I personally haven't done this method, but from what I've read you are only able to cast it once per body and the body must already have been fed on.