Personally, when I returned to League after some silliness IRL, my first games were actually with Gnar in the jungle and that's kinda how I've played him on a number of occasions. He does work and does it incredibly well, especially since he does have amazing base damages, a Q that slows and if you time your passive for a gank, you have a Q to slow them, your W to stun, your E to close gaps, and after your R, you have your R to toss them from that turret, to your turret, or slam them into a wall for the stun. Gnar is unlike actual marksmen though. He can get his jungle item, then a BotRK, and then build pure tank after that and be not only a monster in his own jungle with his huge AoE damage, incredible single target, and his high mobility, but he brings all of that to the lanes and to the enemy jungle, something a lot of ADC's can't do. What a lot of people forget about when it comes to ADC's is their incredible need for items. A jungle can't be an item-hungry burst machine. They have to able to rely on their bases.
Graves can be a monster, but he needs big AD items to accentuate his high ratios and needs the CDR and mana to spam them properly for actual clears.
Cait is a mana-whore and moves far too slow to gank. Out of the question.
Bruiser Vayne is highly possible due to how she scales with both AD and AS. An old friend of mine used to run bruiser AS Vayne and still does from time to time where you would simply get AS items and some tankiness and abuse your passive. BotRK and a Devourer with tank items afterwards would be a murder machine. Her passive would keep her on target, especially when you pop your R. Your R gives you the AD you need, and you have the AS to murder people with your W. Obviously this build is very W reliant and would require you to play very differently.
Ezreal is out of the question since he has two abilities that can help him jungle. His W is completely useless in the jungle. He has no mobility outside of his E and he's more of a poke-champion. He'd be pretty useless in the jungle since he provides nothing.
Quinn is highly viable. Her potency with a Warrior enchant, BotRK, and a LW and then tanky items would sky rocket her through the roof. Her AoE blind, her E that'll stutter the mobs and interrupt attacks, her passive and W boost her damage, and her R gives her incredible ganking potential. She was used as an ADC, Jungler, and Top on release and still does wonderfully on all of these now. She's got CC, a Blind, damage, and can build Bruiser.
Twitch is iffy. He's always been able to jungle. Ever since his release, people have been abusing his stealth as part of jungling to scare the bejeebus out of people with his incredible damage and high mobility. He can work, but his team needs a really tanky top and there has to be CC in every lane, much like what a normal assassin jungle would require.
Jayce in the jungle is not only troll, but it's also hilarious. He really is NOT meant to be here, but for the sake of the Lulz of it all, he needed to be mention. He does have high clear speeds, he can Q+E combo poke as his entrance, Q at the target with his W going in hammer mode and then E them at your partner. It can work, but much like Twitch and other damage-building junglers, you need tanky, CC-based tops and supports to make up for his squishiness.
Corki is a no. Much like Ez, he provides absolutely nothing to a gank, has very limited damage in the jungle, is a mana whore when forced to spam his abilities, and his main item is a Trinity, which is very expensive on a jungler's budget.
Draven can jungle, but why would you jungle Draven? Draven doesn't waste his time hitting mobs. He wants to be admired, to be loved, and steal every single kill. How can he steal the kill if he's gotta take his own red? That's unDravenlike. That's now a word. Patented.
Miss Fortune has her passive to make her camp-to-lane movements quick, but her E doesn't last very long to do much to jungle creeps, her Q has to be angled too perfectly for it to maximize damage, and her gank potential is pretty crap since she lacks CC and won't be clearing her camps very quickly until she's an item or two in.
Tristana is... She's got damage, but due to the nature of her needing levels and her dire need for items to do damage makes her a bit lacking. I could see AP Trist working since those level 6 ganks would be INSANE. Come in from the back, W+E+R them from behind to force them to flash away or die. Her clears would suck though due to her low damage. And yes, I know, her bases are high, but her CD's are HUGE, so her overall DPS is low while her poke is high.
Varus and Kog'Maw move at turtle speeds and are mana whores when they have to spam abilities. Both have poor early games outside of their poking ability and the jungle is unforgiving on squishies.
Jinx could be funny, but her E would be useless outside of ganks. Her damage early is pretty lackluster outside of her poke. Her ganks would also suck due to her lack of mobility like Kog and Varus.
Lucian would be kinda funky. I mean, I would expect a Devourer on him for his passive to abuse. It could work, but his lack of CC would make it a bit hard to actually gank.
Teemo is hilarious. The Goal of Teemo is to take your buffs, then head to their Wolves or Gromp if they started blue, Wraiths if they started Red, and sit there for a moment, wait for them to come take them as they respawn and murder him. I recommend running Blue smite and Ignite to guarantee kills. I also recommend using a vision trinket so you can ward their buffs and sit in rather odd places where they're unlikely to check and wait for them to start the buff. As it gets low, and thus them getting low, blind them, go for the kill. Smite the buff if needed. Try to avoid doing this at red since they can smite red and heal off it and possibly get away because of it. I do not recommend playing Teemo jungle, but if you're wanting to try something hilarious, try out AP Counter Jungle Teemo. Go pure AP. Magus enchant, DFG rush and then what ever fits your fancy after that. Watch your enemies explode and bitch at their laners for not helping them.
Sivir has good clear speeds, amazing poke, an even better initiation, can benefit incredibly well from a Warrior Enchant and has her E to counter possible gank-stoppers like Syndra's E, Ahri's E, etc. She also comes with her innate ability to run at god speed. Like many other CC-less junglers, make sure your team has CC in the lanes.
Kalista is.... weird. She can jungle, especially with her passive, but you'd have to have communication with who ever wants to be thrown at the enemies, which is exactly who you'd be ganking for all game long. If your top is a highly aggressive Renekton, Gnar, or something of that sort who likes to be in the enemy's face and can mix will with you, then I'd say go for that and get fed off top. Her E also works well as a secondary smite. A devourer enchant or rushing a BotRK after choosing your smite will definitely help amplify clear speeds, especially if you level your E first.
Ashe is bad. Her laning phase is already iffy and her clear speeds since she has ONE damage ability on a rather long CD early on, plus her E's passive wouldn't scale well with the jungle. Her R would make for good ganks and timing her passive would be a nice way to start a gank, but her clear speeds would be like Shaco who only has his Q. It would suck.
If I left any out that are considered Marksman, go ahead and mention it and I'll try it out. I've pretty much tried everything I've mentioned. I had far too much time on the PBE while waiting for S5 preseason changes. >>