I pretty much only play solo, and I've got a level 72 zer0 and he is absolutely a blast. I've found that the most efficient way to play him for me was pure melee, especially for farming.
I've found two shield/gun combos to be effective and i'll go over them both.
The first is the law/order combo. You can get both of these for doing easy quests in sanctuary near the end of the main storyline. No killing of enemies required. The order is a shield that gives a pretty large boost with roid damage, and when you melee enemies with the law all damage you do to them also heals you.
The second is the rubi/love thumper combo. You get the rubi as a quest reward from moxxi for turning the "rakkaholics anonymous" quest in to her. the love thumper you'll get as a quest reward in overlook for doing the "best mothers day ever" quest. The best rubi to get is a slag rubi, if you have the time to spend farming the quest to get one. 12% of all damage you do with the rubi returns to you as health, and the love thumper is a pretty strong roid shield that also does explosive damage when depleted.
The reason you'll want either of these combos is because they heal you while you're in combat, and healing is one of the main things missing as a skill from Zer0's skill tree. This means you can get right up in the face of anything, and as long as you can hit them, you can generally let them wail on you without going down.
As for a build, i'd suggest going with this one.
and as you level up, try to work down the cunning tree to get death blossom (if you have the dlc's that let you level up past level 50). many must fall + death blossom in tandem is incredibly powerful.
the main strategy with this build is getting up in the face of what you are fighting, hit it to apply deathmark, let it take down your shield (so you get the roid damage buff and make counter-strike active), go into deception and get behind the enemy (hitting from behind activates ambush/backstab), wait until the deception timer is under 1 second left (to get the max melee damage boost from deception), and then smacking them with execute. it takes a good bit of practice, some set up, and some skill - but once you get the basic concept down... zer0 becomes VERY powerful. especially once you get the health of a major enemy down to ~30% when killing blow kicks in. if your enemy is slagged, you will be hitting them for MILLIONS each melee blow. one more point, is to try to go for critical hits as much as you can as they will basically double the damage you are doing.
if you have any more questions about anything specific - let me know. zer0 is probably one of the strongest solo players in the game, but he does require a little bit of equipment and practice to get that way