>>2793442 (OP)>Is the area outrageously expensive compared to continental Europe?No, but the tourism 'product' is undoubtedly better in continental Europe unless you have a specific interest, like industrial history, popular music history, or football stadiums. Mancs and Scouse are both quite friendly to strangers, in my experience.
>What would I need to avoid in the area and what would you recommend?I recommend one of the many local museums dedicated to industrial heritage, but maybe that's just an interest of mine. Both Liverpool and Manchester are pretty nice (or at least, first world) in their centres these days. As you say, Chester is a must. Lake District is not too far away, but you might not get much out of it in such a short time, unless you spend the whole holiday there, which wouldn't be a bad idea. York is a fantastic holiday destination that you can easily get to from Manchester, either to stay or visit. Check out the railway museum while you're there. Manchester also has an excellent transport museum.
Some suburbs of both cities are still very rough. If you're an American then you will be aware of the concept of post-industrial wastelands full of the terminally under-employed, though we also have seething resentful thirdies thrown into the bargain. Also best to avoid the satellite towns of Manchester - Oldham, Bolton, Wigan, etc.