To slightly expand on what the previous comment is saying, I always refer to a "successful jungler" as someone who can efficiently keep up a par time on their first and second clear and gank(s). Beyond this point, ANY champion can clear jungle camps and gank to some degree of efficiency with items that suit their kit respectively.
Stronger junglers tend to have built in sustain, AOEs, multi target damage, resistances, damage reduction, attack speed buffs, auto attack resets, and/or kiting abilities.
There are a lot of champions who have these, and so, there are a lot of champions who are able to jungle to some degree of effectiveness.
Also as the other person mentioned, a very important factor to consider is how vulnerable you will be to invasions while attempting these first few clears.