TL,DR: In general, if you're looking for champs, look for CC, survivability/tankiness, and escape or mobility. Ranged often trumps melee because of poking ability, but not always, especially if the melee has a gap-closer such as Jarvan IV or Xin Zhao. You need to look at how the kit works together. Ryze, though very squishy, can become super tanky with items like Frozen Heart and Odyn's Veil, both naturally in his core build. He has sustain with his ult, hard CC, and that CC can often be enough to let him get away.
Also, bot lane is a whole different enchilada. Playing bot can win you a LOT of games if you learn to do it well. Generally, any champ you'd put solo mid or solo top in SR can make a good bot champ.
Many champs that aren't actually bad require good teamwork to do well. For instance, Ashe-- she needs a tank to protect her, and 1v1 is generally very squishy. She can do extremely well, as long as she sticks around with a tank.
Basically, consider all of Dom to be one big SR jungle. Since in Solo Queue, you might not even get a good team COMP, let alone a good team, any champ that's strong in jungling can (generally) be considered strong in Dominion.
Second, because you can't rely on having your whole team there in a fight, a champ with escape mechanisms is also strong. Suppose you can 1v1 any other player in the game, you're still gonna find it harder when they come at you in twos or threes. Having tools to escape that fight makes you a stronger champ, since those kinds of odds happen frequently. Mobility can be a good, or even better substitute for escape. It means you can prevent caps, join teamfights to improve the odds for your team, and still usually escape from a gank.
Third, CC is very strong. Not only can you keep people on your turret taking turret hits, but you can take people off their turrets and kill them, and contribute more to a teamfight. And again, CC can frequently let you get away.
These three points combined is why there is a new "Rammus OP!" thread every week. He's a strong jungler, he can take damage, he can escape, and he can CC. Shaco is similar; hard CC, good escape, and can often do well 1v1.
Fiddlesticks is... a special case. He's a great jungler. He has hard CC. And his damage is also a good sustain ability, so 1v1 he often doesn't need an escape, because he can usually handle it. Problem is, one gank and he's dead, because he can't do anything while channelling his drain, his silence wears off fairly fast, so does the fear. So being exceptionally squishy, anything more than one enemy means he's probably dead. Fiddle can do great in teamfights. He can do great solo bot lane if he's properly wary of ganks. But in general, not the best soloQ champ.
So Warwick, I've seen do pretty well. Lifesteal is gimped on Dom, but he's good 1v1, and he has a CC in his ult. His chasing ability isn't really that good since chasing kills is usually a bad thing, and he can't escape if he gets ganked. He does terrific in teamfights, and if you build him tanky he can do pretty good solo. He's not a trash champ at all. Not the best, but not a trash champ.
Veigar is boss in teamfights. Having a 1:1 AP ratio nuke drop on 2-3 champs at once will ruin anyone's day, and his Event Horizon is an AOE stun. But by himself, he's super squishy, even with his stun, and he can't farm minions all day for his Q to get majorly powered up. Also, because of the Dom Armor Penetration aura, you tend to see more AD champs, which means his Ult is nerfed too. . Unless he's in a teamfight, he's generally pretty bad, and you can't guarantee teamfights in Dom.
Ashe does beastly damage. But 1v1, even her slow might not be enough to prevent the enemy getting to you. Doubly so if you get surprised. Her Ult can be good to prevent caps, but she also has no escape mechanism aside from that. If she has a team to back her, she can do terrific. Otherwise, she will most likely be in danger.
Nunu is very tanky, has lots of slow, does lots of damage... but his Q only works on minions, which are not in the jungle, and if he takes bot lane, he has no real AoE to push the lane. If 1/4 of your kit is practically useless, then there's very little point in playing him. He *can* do well, especially in a team that can make full use of his abilities... but there, again, is that Teamwork word that you can't guarantee is there in Dom.
Teemo can be very strong. Build him right, play him right, and he will absolutely WRECK the enemy team. He has a couple of build options available to him, he's squishy but can escape fairly well, he has perma-stealth to prevent being ganked, and his Blind, though a soft CC, is good both for him and for teamfights. His shrooms are good scouts and can do lots of damage in and of themselves with the right build. He can soloQ extremely well. A player who does not play him well will die hard, but a good player can wreck face.
Garen... I've never played as him, but I've played against him, and man he is annoying. Silence, spin, tanky, damage... hard to get him off that point, will die to ganks but so does everyone. Maybe someone else's mileage will vary, but I would say he's at least decent, not weak.
Shen: I have seen exactly one dominating Shen. All other Shens I've seen just can't cut the mustard. The Shen I saw built his defence right, Ulted into the right fights, Ulted away when he needed to escape, was always in your face, and he has an AoE taunt. But he doesn't really do much damage, has no real range, his escape is an Ult which isn't instant, and while tanky just doesn't seem to do well.
Champs I haven't seen much of include Rumble, Corki, and Trundle. I know Corki can put out lots of damage and has an escape in his Valkyrie, so I'm not sure why we don't see him more often. Trundle has no hard CC, little sustainability, can be tanky but his escape is only marginally better than Kog'Maw's. Rumble... I'll be honest, I know next to nothing about him, except that Heat isn't buffed by the Dominion aura.
Many of the supports that needed a team to win can soloQ and do well if built selfishly instead of to support the team; both because you don't need to rely on your team (which might not be that good) and because you get farmed automatically. Soraka can bot lane pretty well, Janna is fast, can shield, ult/knockback, heal, and put out some decent damage. AD Sona I hear can do every well too. Kayle can do lots and lots of damage, has decent range, and has great survivability in her ult and in her usually tanky build.