How long does your current lease have to run? In your situation I would be saying to the roommate, "If things do not improve and stay improved, when the lease is up for renewal, we will be expecting you to find somewhere else to live." If he continues to be a pig, have the landlord redraw the lease with your names, and line up a new roommate to take his place. I assume this can be done fairly easily because you state you have had other roommates and that you all have to be on the lease, so presumably it wasn't too hard to get them on and off the lease.
If the roommate suggests that YOU be the ones to leave, point out that the landlord is not going to choose a tenant whose housekeeping style invites rodent infestation over someone who cleans. Tell the landlord this if necessary.
I think you will have to steel yourself to cleaning his room after he leaves, because if he hasn't done it to date, I doubt he will do it after being essentially kicked out. Perhaps you can retain his portion of the renter's bond as recompense.

First of all, I agree with colleenoz; the chances that he will clean up after himself when he moves out are negligible. If you really want to stay there, brace yourselves for cleaning up his mess when he's gone. (Actually, even if you don't want to stay there. If you all move out, you can and probably will be held responsible for his mess--see below about jointly and severally--, so you might as well accept that you'll get stuck cleaning.)

As for getting him to move, I'm assuming you've asked him to move already and he said no? That would be the first step, if you haven't done it. If he agrees, then you talk to landlord about signing a new lease without him. Which, like colleen said, it sounds like something that's acceptable to landlord if you've gone through 5 roomies in 2 years...

As for getting him off your lease if he doesn't want to go, you probably don't have a lot of leverage, but it depends on your relationship with your landlord. If he is on the lease, the landlord is obligated to let him live there until the lease is over. That said, most leases require tenants to maintain the place in accordance with basic health requirements, which it sounds like he may not be doing, which would give the landlord (not you) a legal recourse for kicking him out early. However, most roommate-situation leases are also written "jointly and severally", which means you are all responsible for each other's behavior; if the apartment is bad enough that your landlord can kick out the roommate, he can also kick you out. Doesn't matter whose mess it is (or who hasn't paid their share of the rent, or who's making all the noise at night, etc). And it might be easier for him to just get rid of you all than allow himself to be dragged into the middle of a roommate dispute. But, if you have a friendly relationship with the landlord, you might be able to get him to kick out roommate only.

Or, you can forget trying to get rid of roommate on lease violations and just wait until the lease is up. Then, you can move. You can stay, and ask the roommate to move, but you can't force him to. (your landlord can) Once the lease is up, your landlord could choose to rent to you, to your roommate, or to none of you and just find a new tenant. Again, the landlord may be inclined to just get rid of you all rather than put himself in the middle of your problems, or if you are friendly with him, he may be willing to renew with you but not the roommate. (or if roomie is friendly with him, he may do the reverse, although that seems unlikely if your description of his hygiene is accurate).

This topic is probably long since settled, but fwiw:

Mediation. Many communities offer it at a very low price. If yours doesn't, it's still a good investment.