Easy if you buy the same type of lock, usually only a couple of screws. you might be best to take it out and draw round it then take that to B&Q or somewhere so you can get one that fits in the same space.
Don't know how to phrase this without sounding stupid but it's a lever type door handle, with a traditional looking key - hoop on the end, stick then the key bit (not a yale).
Would I need to get a new door handle, or can I take the old one off and just replace the lock underneath?
No, you shouldn't have to replace the door handle - you should be able to use your current one.
If you've got a digital camera and printer, take the lock mechanism out, take photos of it and print these so you can take them to the shop to compare with the locks on offer.