As a fellow Melee Krieg, if you want to go pure melee, just go in the Bloodlust tree; if you want to go hybrid, go in the Hellborn tree to get Delusional Damage and go back into the Bloodlust;
As you can notice from this build I’d suggest going in the Bloodlust tree, so you can continue to melee everything with BXR (remember Release the Beast) and Buzz Axe Bombardier (and later, Bloodsplosion); just get some slagging methods (a good bladed Slagga/Grog/Rubi or some slag transfusion grenades/Magic Missiles), learn to throw your axe when you can’t just melee everything (Badasses and Constructors come to mind) and remember that throwing axes won’t trigger Silence the Voices; learn to throw your axe because you’ll deal a lot of pain with the Bloodlust tree (remember to trigger Fuel the Blood first if possibile because it’s boosting your throw’d axe damage, but you can switch your points to get Blood Overdrive too; pick your poison), and learn to distance yourself in need.
For the gear:
For weapons, you really only need some Torgue/Bandit with Torgue barrels AR weapons for getting yourself in Release the Beast range (never do the mistake of wasting BXR without Release the Beast except for real emergencies); some other explosive weapons (Swordsplosion? Unkempt Harold?) because they’re boosted by half your skills, a Moxxi weapon for healing (not mandatory but really appreciated), and some slagging guns (Rubi/Grog works wonders being Moxxi weapons AND slagging tools!)
For the shield you can go with the Love Thumper, another good roid shield (Law + Order combo is good to have when you aren’t BXRing, and the blade actually boosts your damage even in BXR, but every roid shield with a lot of recharge delay is good for Salt The Wound) because they will boost your damage to insane levels and FotF is just plain garbage for Krieg without points in the Hellborn tree; the other way for Krieg is basically the Rough Rider (bonus health and a 20% flat damage reduction which works on your selfhit from Silence the Voices too!), stop.
For the grenade you can go with a (preferably Slag) Transfusion if you aren’t using Moxxi guns, or just Magic Missile for slagging everything (if you can’t get one, just go with a low level Slag Bouncing Betty one); all of them preferably Longbow with a low fuse
A Sickle class mod is almost mandatory (preferably a blue one with +6 Silence the Voices and +5 Empty the Rage, but every Sickle works wonders); you can get a Slab one if you are early enough and still in TVHM but you’ll probably need to switch for a Sickle ASAP.
Explosive relics are your ticket to heaven in terms of damage, Vitality relics are your bulkiest call; choose your poison.
Like I said, always remember to play smart with your axes; throw them if you can, go Release the Beast everytime you can and destroy everyone in your path!