It's a lovely place to live, we moved there from the south a few years back. Although it looks like a quaint old city there's actually loads going on. E.g. A fab new cinema and theatre, exhibitions, music, brilliant restaurants etc.
Nice areas to live are probably Hoole or Handbridge if you want somewhere cool with its own stuff to do/ high street but within walking distance of town.
Or Westminster Park, Upton, Newton, vicars cross if you want somewhere a bit more residential or a bigger house with a garden.
Broughton (near Waitrose) is up an coming but does have a lot of rental properties.
Avoid the Garden Quarter if you don't want to be surrounded by students! This is the roads round the uni and canal like Cheney road, Catherine street etc.
Blacon , Lache and Saltney are the most 'deprived'areas but nothing like London deprived areas, they're fine!
You could also consider the villages as they are so near- mollington, christleton and Waverton are lovely.
I cannot think of any primary schools to avoid, we're so lucky to have excellent schooling here however many schools are full if you're planning on an in year transfer.
Secondary- christleton, Bishops and Upton are most popular and get good reviews. Christleton and Upton look like they get better exam results but Bishops is smaller with great pastoral care. Queens Park has had a bad few years under a terrible head but the new head is great and turning it round, still only 60 kids started Y7 this year when it has capacity for 180.
Beware that high schools give some priority to feeder primary pupils so if you have a strong preference for a particular high school you may want to apply for a feeder primary. The only oversubscribed secondary this year was Christleton though.
Wirral (neighboring council) have grammar schools so chester kids can take the 11+ and apply there if they don't mind traveling a fair way to school each day!
Hope that helps, just ask/ PM me if you need more info!!