First a freewheel and a cassette are two different things. Given the age of your bike you are most likely to have a freewheel. You can clean and rebuild them but its fairly fiddly to strip them down as they are full of small bearings. Its probably going to be much easier to replace the freewheel with a new one. You will need a freewheel removal tool depending on the make of freewheel you have. Some freewheel tools are splined, some have pegs which lock into the freewheel for removal. You could ask your LBS to remove it.
If you currently have 27" wheels changing to 700c will change the caliper reach for your brakes, so you may end up having to replace the calipers as well. If you check out Ebay it should be easy enough to find a pair of alloy 27" wheels. Its also a good place to find a new freewheel and any other items you may need