尽量 means do the best he/she can(having the ability to do something but only trying to find the will to do it). For example, "我尽量少说话。" ("I'll try to speak less", you are able to decide how much you speak but you only try to speak less)
尽力 means do the best he/she can(not having the ability to ensure it but try one's best to work on it) For example, "我尽力在下周一前把这个项目做完。"("I'll try my best to finish this project by next Monday", you don't have the ability to ensure it, but you'll work on it)
尽可能 means as much as possible. If you are not sure when you should use 尽量 or 尽力, then use 尽可能 to save yourself some trouble. Though nowadays 尽量 and 尽力 seem a bit interchangeable and people generally don't bother to distinguish them. Still there is a difference when talking to your Chinese boss(if you have one, perhaps in the future), with 尽量 you sound much lazier but with 尽力 you sound much more determined.