Aintree is miles away from the city centre. There's nowhere really that close to the station.
There's Scarlets (never been there) in Widnes, which is the other side of the Mersey from Runcorn. Unfortunately if you've only got trains to rely on then it's a PITA to get to from Chester. I agree with the others. If you're prepared to get a train out of Chester to punt, go to Manchester.