So I just hit lvl 50 and cleared TVHM with a melee Krieg, total playtime is 1d 13h.
He's by FAR the easiest and most powerful character to hit 50 and clear TVHM with. I was meleeing things for over half a million damage each time.
After you hit lvl 31 and reach the bottom of the mania tree he becomes unstoppable, I was one or two shotting most enemies.
The final skill of the mania tree is way too good, instant full health recovery, +50% damage reduction, action skill completely recharges upon finishing. The best thing about it was that getting killed halfway through and getting a 2nd wind from FFYL also gives complete action skill recharge.
Not the most fun way to experience the game for someone just starting out in BL2, but I've basically done everything else, so I really enjoyed it.
Equipment used:
Early game: Law + order (+100% melee, damage becomes health recovery)
After normal mode: Rapier from pirate DLC (+200% melee)
Roid shield for the entire game. Doesn't have to be love thumper.
Class mod that gives + melee damage
Artifact that gives +health
Skill progression:
Empty the rage, Fuel the rampage, Thrill of the kill, Salt the wound, Silence the voices, Release the beast.
Once Release the beast is obtained swap out Fuel the rampage with Feed the Meat.
After lvl 31 the choice is really yours, but I prefer the Bloodlust tree.
Blood twitch, Taste of blood, Buzz axe bombardier (god tier skill), Fuel the blood, Blood trance.