Proverb or expression for someone taking on too much

For doing something that is too hard, or doing too much of it at one time, I'd say you could go with

Biting off more than he/she can chew

which usually implies "taking on too much/too may tasks", but I think it's fair to stretch it to "taking on a task that's too hard for you". This is, to me, the most obvious idiom for this, but it's fairly cliche.

You could also use

Getting in over his/her head (as in, the water is too deep)

which implies swimming somewhere where your feet can't reach the bottom, and yet you're not a strong enough swimmer to handle it. You can also use

He's out of his depth/He's in too deep

which is the same metaphor. Neither of these imply failure, since the speaker may have misjudged the swimmer/chewer. I can't think of a phrase that combines these with the notion that the task is impossible; usually impossible tasks are impossible for everyone and not just for one person (swimming in deep water may be impossible for him, in which case he's getting in over his head, but it's not impossible for me, in which case I just swim there).

I really like Mr. Shiny's answer, and I almost just made this a comment there, but I suppose it deserves its own answer.

You can say that someone is engaged in a Herculean task. When used alone, the word herculean implies great strength, but an Herculean task connotes "requiring tremendous effort, strength, etc.", in reference to the seemingly impossible labors that were required of Hercules in Greek Mythology.

Because Hercules succeeded in his labors, this idiom wouldn't go quite as far as to say that the task is utterly impossible--only that you'd have to be on par with a demigod to accomplish it.

Update

Along the same lines, but with a stronger emphasis on failure, the American Heritage Cultural Dictionary has the following note on the definition for Sisyphus:

A difficult and futile endeavor may be called a “labor of Sisyphus” or a “Sisyphean task.”

Either of these phrases requires at least a cursory knowledge of Greek Mythology, so they may not be appropriate for your audience, but saying that someone is engaged in a Sisyphean task seems to match your criteria well.

The only one not yet mentioned:

Too much on her plate.

Implies overcommitment and a lack of spare time.

You have listed very specific criteria, and unfortunately, I doubt if there is actually one single idiom to describe all of that. One could possibly use a combination.

For "doing too much" :

Burn the candle at both ends.
Bite off more than he can chew
Carrying it too far.

For "illegal" :

You'll be caught red-handed
Taking the law into his own hands.
Fool's errand

For "impossible":

Flogging a dead horse. Nailing jelly to a wall
Mission impossible

I would suggest, if you were to want a sentence expressing all three of your criteria, something like this:

You can't catch lightning in a bottle(impossible), besides, you'd probably be caught red-handed(illegal), and end up trying to burn the candle at both ends(do too much)