I'm starting a new character. I'm not really sure what would be the best to use.
There is none who is "made" for playing solo. Every character in Borderlands 2 can be played solo. They just have different fighting styles and therefore, some are better for soloing the game. Note that this only applies to the first 2 difficulties. Once you hit UHVM. The game will become harder. MUCH harder.
Tactical fighter, uses mainly mid- or long-range combat weapons like assault rifles. Action skill is a turret which can deal huge amounts of damage or grant shield/ammo. Excels at extended fights or fights in narrow rooms. He is literally a walking tank if played with the rough rider and the skill crisis management.
Personal rating: Probably one of the best characters for playing solo.
Assassinlike, can lock single enemies or deal huge damage to certain enemies (e.g. bosses). Prefers SMGs. Focussing on the right tree (elemental damage) can destroy waves of enemies within seconds, but lacking any valuable possibility to keep bosses at bay (e.g. terramorphous). Also, a rough rider build on her is completely overpowered and makes her almost unkillable.
Personal rating: Good to play solo, but can become difficult in UHVM and farming legendaries with maya is not that easy.
Pure assassin, focuses on stealth, sniper and critical hits. Therefore, I don't have to mention that this guy is awesome in solo play. Unfortunately, snipers become futile in UHVM or above, so you can't play like that forever.
Personal rating: Good to play solo, but it's necessary to switch weapons and/or skilltrees in UHVM.
The rambo of the game. Standing out in the open with 2 weapons and shooting everything. While this may be easy in normal- and TVHM, this strategy will become almost impossible to play if you are not completely overequipped. You will die - a LOT. Salvador is best in a team where he can draw attention and get revived in emergencies.
Personal rating: Probably the worst character to play solo, except you use bugs
DT OP. I don't have to say anything else. Once you hit 5, the game isn't a challenge anymore except for some bosses. It becomes challenging again once you enter UHVM.
Personal rating: It's not a problem to beat the first 2 playthroughs without dying once.
Well, krieg is special. While many people may disagree with me now, saying that you get so much health back/dmg resistance while you burn that it's almost impossible to die, this was the hardest character to play through the game for me. I died more with krieg than with any other character. Maybe it's just because I suck with krieg (I hate his entire playstyle), but this character was purely made for teaming up with others.
Personal rating: NOT made for solo play in my opinion. Needs backup of a good team or at least a heal siren.
With the caveat that I've mostly played solo so far:
I played my first game with Gaige and found she was a great solo character, especially pairing the Deathtrap with an elemental sniper rifle. I've definitely died more often than once though. :)
Right now I'm entering Eridium Blight on my first Siren playthrough, and using the Cataclysm and Motion trees (Converge + Chain Reaction). I got lucky with a Thunderball Fists drop in the Preserve, and that's been a very powerful means of dealing with WAR and badass loaders, especially when phaselocked. Barring a legendary, shotguns and scoped weapons make good phaselock pairings because you can crit more easily.