Is there a viable way of completing Ultimate Vault Hunter mode by yourself?

The most standard way of playing melee Zer0 is to go all the way down the Bloodshed tree for Many Must Fall. Definitely skip Grim because it ruins how you will be playing and Resurgence is useless with the weapons you will be using. See this post for the complete melee formula (in short melee damage is based purely on your level and other modifiers, weapon damage is irrelevant unless it specifically says it does extra melee damage) - you can use this to min/max the skills. Many Must Fall is better than Death Bl0ss0m typically unless you are doing raid bosses so switch if you are farming raid bosses.

Your standard setup includes:

• Rubi (preferably with Slag and Evisceration)
  • An alternative is to use Law with Order but I believe you will do less damage overall
  • This is how you are going to recover HP - ~12% damage you do with Rubi is gained as HP, you will often go from 1% to 100% HP in one hit
  • Any damage done while holding Rubi, not just damage done with Rubi is recovered as HP
• Love Thumper (preferably with one of the immunities)
  • This is the better alternative to Order for the damage that it produces which means better HP recovery
  • The reason Love Thumper is better than Order is because you don't need to manage your shield and the higher Roid damage
  • Another alternative is Hide of Terramorphous but again you will need to manage your sheilds
• Rapier (also preferably with Slag)
  • the 200% modifier is huge and if you look around you can find videos of Zer0 doing max damage in one melee swing
  • Swap and use this as necessary - DO NOT underestimate this weapon, this is a huge boost to your melee damage compared to Rubi
  • This requires Captain Scarlett and Her Pirate's Booty DLC which is pretty good overall
• Proficiency Relic is the most common relic used
• Class mod is up to your preference
• Storm Front
  • Often used because it has consistent damage so paired with Rubi it has good HP recovery and the explosion can break your shield while the Tesla won't damage you
• I usually keep a Rocket Launcher, Sniper Rifle, or Dahlminator in the remaining slots for FFYL, long distance, or flying enemies

Remember Roid damage is a big contributor to your overall damage so your shield should usually be depleted - this is why Grim is useless and detrimental in this build.

With Rubi in hand you can pretty much just trade blows with most things. If you are using something similar to this build, your damage output in UVHM will be "sufficient" as well as your sustainability.

If anyone tells you Zer0 is not viable, they are either not using something like this or do not know what they are talking about. As far as I know Salvador is the only one that comes closest/surpasses Zer0 in damage output. Gaige's Deathtrap and Axton's turret(s) don't come anywhere close to what melee Zer0 can dish out.

Yes, this does mean pretty much every gun you get is near useless and their damage is irrelevant (remember you still have 2 slots you need to fill). The only values that matter in term of damage is your level, melee modifier, and skills that modifies melee damage. As far as I know, nothing of value was added in UVHM for Zer0 players.

More info organized here.

Some interesting video:

• Max damage done in a single hit - I believe it is because Death Mark from multiple players can stack
  • UVHM Voracidous The Invincible Melee Solo
  • Pyro Pete solo - PT2.5 I believe
  • Pretty much most video by Gothalion
  • Alternative to melee Zer0 using B0re on Hyperius

I am currently doing solo UVHM as Hybrid Zero, and it's been a breeze. Slag -> correct elemental weapon—even badasses and super badasses go down in few hits.

(I don't really buy that "getting killed in 1-2 hits. My Zer0 has 400k health and is perfectly capable of soaking damage.)

Although I do confess that the constant weapon swapping gets really tiring after a while, it's so routine and tedious:
shielded bandit -> slag -> shock -> fire -> reload all weapons -> next enemy-> rinse and repeat ad nauseam...

But I wouldn't say it's hard. Just use those elemental weapons and remember to slag the hell out of everything, and a few well placed shots are all you need to take down the baddies.

For melee builds, use slag singularity grenades before going for Many Must Fall, those executions deal insane amounts of damage on slagged enemies.
Kunai also benefits from any bonus for "melee damage" since it registers as a melee.

B0re can totally devastate groups of enemies. Just drag them together with singularity slag and fire away. If done right, everyone is dead before your magazine is empty.

If you have Tiny Tina's Dragon Keep DLC, then go find a Magic Missle grenade mod dropped by a wizard. Regardless of what level you find one at, they all do the same thing.

1) they regenerate your grenade ammo 2) they fire 2 homing nades that home in on opponents 3) the homing nades have an almost 100% chance to slag on explosion

Toss a Magic Missle. Let it slag some folks while you switch to the appropriate elemental sniper (fire for flesh, corrosive for armor (yellow health bar), shock for shields, or explosive for all-around decent damage on everything). Let the homing Magic Missles do their trick.

This ensures you always get slagged enemies, and then you pick them off. If you team it with something like the Pimpernal from Scarlett's Pirate Booty DLC you can do some major damage. Problem with UVHM is that levelling scales even more exponentially, so weapons that are 2-3 levels behind get useless very fast. It's better to just move on to a same-level white gun than to stick with a lower-level green, blue, etc.