The proper way to deal with a chain is to first take it off the bike. An old, worn out chain hooked to the one in use makes this an easy task. Once the chain is off, clean it thoroughly with paraffin or petrol or some similar solvent. The chain can then be laid flat and the ends pulled and pushed to see how much wear there is.
The next step, provided the chain is not worn out, is to heat a tin of Linklyfe
graphited grease until the grease is well melted. This can of course be done on
a cooking stove, but to avoid the smell in the kitchen, a camping stove outside
works very well. The chain can then be put in the Linklyfe which will penetrate
all moving parts. When cold, remove the chain and wipe off surplus grease.
Put the chain back on the bike, adjust it, and you're good to go.
Provided chains are replaced before they are totally worn out, there is no need
to replace the sprockets. I do know Japanese bike dealers always insist on
replacing both chain and sprockets, but I strongly suspect the reason for this is
more financial than mechanical.
Colin.