Very soloable. Some classes have a slower start than others, although the P2W vendor bonus powers (Sands/Axe, Nemesis Staff, Blackwand) help a lot in filling out your early attacks for any archetype. The Prestige Enhancements help a lot too.
Typically a lot of people who solo games, do so because it's just not efficient for them to spend the time looking to team up with someone, deciding what they are going to do, getting everyone there, waiting for everyone to be ready, etc. etc.
One of the 'magic' things about CoH, IMO, was always the way that you could just jump on, type a quick 'Lvl 6 Blaster LFG' or even just say 'DFB forming pst for invite' and boom, 2 minutes later you have a group, people are teleporting/superspeeding to the entrance, the group slams in, starts to roll the content. Doesn't matter whether you have blasters or tanks or defenders or whatever, except in rare cases any team composition tends to work.
Team not working out? Thank them for the run, drop team, pop out another LFG. Boom 3 minutes later you're on another team. I often felt that CoH was the game where you could say "we're soloing fast, together". :)
As others have mentioned, I often solo because I want to knock of some contacts and missions, and teaming just levels too fast. So just start a character that amuses you, give it a few levels, if it isn't working out try another. I generally find that in CoH, it is more important to decide how I enjoy playing, than to plan for solo/team play.
- Sometimes I just want to hammer whole groups. AoE and Control and AoE debuffs - Blasters, Sentinels, Controllers/Doms, Defenders can do this (with the correct power set choices)
- Maybe I want to just slamhammer single targets, BOOM you're down! Brutes, Scrappers, again Blasters though trickier.
- Sometimes I feel like just neutralizing entire groups of foes while I decide what to do with them. Controllers/Doms again, some Defenders (Dark/Dark), some Sentinels maybe.
- Other times I just want that earthshaking KABOOM of my powers. Superstrength, most Earth sets, maybe some of the newer melee sets (I haven't tried them all).
Find the style that entertains you just to be doing it, and you will soon find that solo, team, simply running around and helping other people without even teaming... it's all fun.