Buying a PC or laptop off the shelf is crap, you're always paying way over the odds.

Based on what you paid for your original PC something like this spec wise would see you right, although how they class that as a basic setup I dont know. It's a small form factor so if you're into gaming or your kids are and they want to stick a decent video card in it you're out of luck, a full tower would be your best bet.

LENOVO IdeaCentre 510S Desktop PC Deals | PC World

Again it's a mechanical drive, an SSD would be better.

For comparison. I just bought an i7 desktop off some muppet on gumtree, it came with 4gb RAM and a 500gb hard disk. I paid 30 quid for 16gb RAM and sold on the 4gb dimm for 15. I had a 120gb SSD already, I bought 2 for 25 quid. Its now a proper rapid workhorse that owes me 130 quid. It's not pretty, it's an ex corporate machine, but for what I want its a steal.

If you want to buy a new PC thats cool, but if you're trying to keep costs down I'd do this.

Stick with your old PC for now. Get it cleaned up, new paste on the chip. go to crucial.com and run the memory tool, see what ram you need and then go buy that ram from ebay, 8gb will do you. Pick up a cheap SSD, every Tom Dick and Harry are selling them for 20 quid a pop now (for 120gb). Install a moody copy of Windows on your SSD and use your existing disk as a data drive. You could do all that for around about 100 quid. the only money out of that 100 quid you wouldnt get back is what you spend on getting it cleaned up, the memory and SSD will hold its value if you sell it on in the not too distant future. You could do the PC clean up yourself, it's not rocket science.