All classes can solo in DAOC. Some just do it better than others and a few do it excellently. I'm most familiar with Albion classes, so I'll describe them. I'm sure someone will be along to tell you about Hibernia and Midgard.
The stealther classes - scout, minstrel and infiltrator tend to be the most difficult to solo. There are templates to make them decent soloers, but it seems to keep getting more difficult the higher they get. Scouts and infiltrators make good RvR classes. Minstrels are most welcome in groups.
The next most difficult classes are the tanks. The Armsman (heavy) and Mercenary (light) have no spells and have a longer recovery time than most other classes. They can solo, but not quickly because of down time.
Clerics cannot train a weapon so they have sorry melee skills. If they spec in their offensive line (smite), their healing and buff lines will suffer and make them of limited value in groups.
Friars and Paladins are good classes to solo. Friars are the secondary healers of Albion and get to train their weapon - the staff. Although they can't replace a cleric, depending on the makeup of the group, they can do a fine job in certain circumstances. Friars only wear leather so although they can tear up an opponent one-on-one, they don't last long when beset by more than one attacker. The Paladin is a heavy tank with a lot of utility chants. Their weapon skill is less powerful than the armsman, but their chants do wonders in keeping them alive.
Next come the traditional casters - wizard, theurgist, sorcerer and cabalist. All can solo well under ideal circumstances. All of them need to be able to drop their opponents before they get within melee distance because most things, especially at higher levels, can one-hit a cloth-wearing caster. Casters basically need space and sparsely distributed mobs, however casters, when they are doing it right, can take out multiple mobs at once.
Finally, we have the newer classes. The necromancer is, hands down, the best soloer in Albion. I find it an extremely boring class, but you may like it. A lot of players raise a necro to pharm cash and other items. The reaver is a fun class that can solo well. It is a hybrid caster/meleer and is pretty good at it. The newest class, coming in with Catacombs is the Heretic. It seems to be a very viable class, but like the traditional casters, needs space to work well.
I suggest going to classesofcamelot.com for concise information on classes, abilities and creation/play guides. There is some good information here, but CoC specializes in just that information.