I have a great read on my opponent. He's 13 vp$ip 3pfr. He seems to be making a killing at the $100's as he's always sitting on double stacks when I show up. I'm pretty sure he's a set miner.

UTG+1 calls (posting dead)
MP1 calls
Hero's dealt A[d]K[c] in the CO and raises $5
Folds around
MP1 calls

Flop ($12.50)
A[c]6[h]8[d]
MP1bets $7
Hero raises to $20
MP1 calls

Turn ($52.50)
A[s]
MP1 check
Hero??

Should I check behind and hope for a miracle river, or should I bet here? Does anyone think I'm ahead?

Based on those stats, he has AK / AQ or a set. He doesnt have AT or anything like that. If you have a "great" read on him you should know if he leads with a set or not.

Against this guy I really dont see any reason to raise the flop.

We both had fairly deep stacks at this point. I think it was $150 for me and over $200 for him. Also, I raised his flop bet because he's the type of player to bet at a flopped Ace trying to steal.

Once I raised and he called I put him on a set. I checked behind on the turn. A king came on the river and he led for $35. I raised him to $90. He called and showed 88.

Flop: Call
Turn: Check
River: Push

If you are fairly confident he has a set, I don't see any reason to raise the flop, especially with the deeper stacks. The turn check is good, giving you a chance to spike the miracle card, which you did. Once you hit you card on the river, you need to get all-in and stack this guy.