I personally like playing a Hellborn Krieg, even the accepted norm is melee.
Here's a sample Hellborn build. I enjoy it with the Legendary Reaper COM because I like Blood Bath too much. If you are capped at level 61, simply don't put any points into the Mania tree. The key skill is Elemental Empathy--I don't use the FotFH/Grog combo because that's just crutch. Healing is quite easy when you have EE, and your damage potential is just absolutely insane with Blood Bath enabling you to one-shot every common enemy. The synergy with those two skills is also insane--since your damage is off the charts, your DoT will similarly instantly heal you.
If you prefer a melee build, here's a decent one. It could be argued that the 2 points in Boiling Blood be swapped to something else, of course. You've once again got the best skills Krieg can offer for melee. Bloodsplosion will be boosted by Fuel the Blood and BxB, which will greatly increase your damage potential. A hybrid melee/Hellborn build is also possible, but it's a little weird and you'll probably be missing your Bloodlust skills.
For gear, you'll pretty much always want a Rough Rider shield. Have the FotFH/Grog combo in your pack for those situations where you need the constant healing (bosses, raids, heavily populated areas like the Slaughterdomes and Sawtooth Cauldron). For relics, I like the Blood of the Ancients, with two ammo capacities of your choice. The Legendary Sickle and Reaper should be your go-to COMs for melee and gun, respectively. Finally, the Storm Front is the standard grenade mod, but I also highly enjoy a slag Bouncing Bonny.
As for guns, get weapons that will proc BB. The DPUH, while recommended to high heaven, is still an effective and extremely powerful weapon with Krieg. I've also been enjoying the Omen shotgun, although that might be a little harder to acquire.