I've done the opposite to you - born and bred Chester, now live in London!
I grew up in a village about 20 mins outside Chester (in fact, it sounds suspiciously like where SuperMonkey lives!) I went to the local primary and secondary schools, and walked to school every day. I suppose it's 12 years since I left, but I do remember it feeling very safe.
Chester is a great city. The only reason I wouldn't live there is because it's where I grew up, if you know what I mean. It has all the advantages of a big city (great shopping, bars and restaurants) while being small enough to walk around, and retaining lots of character. It's a really beautiful place, and the river is wonderful in the summer.
As far as schools go, there are some very good state ones (Tarporley and Christleton High were very good when I was there), or Kings if you can afford to go private.
I don't know what you would consider "very northern"! I claim to be a northerner, but down here people tell me that Chester isn't in the north (despite it being on a similar latitude to Sheffield!) All in all, I'd say that no, you wouldn't find it very northern. Accent is like a milder cross between Liverpool and Manchester.