1st off it's never a good idea to just change either the chain or the sprockets on their own. These should be changed at the same time as each other, hence the reason you get a lot of chain and sprocket kits.
The reason behind this is that the chain and sprockets wear in together. If you put a new chain on old sprockets that chain will wear down a lot faster and visa versa.
To remove the old chain, the quickest and easiest way is to angle grind the old one off. It's a good idea to do this with the rear wheel removed as not to damage it or the tyre. And is easy to cut where the chain hangs below the swingarm. The new chain will be already split with a link to be used to join the chain back up. This is where the chain splitter/riveter comes into play. IMO all the cheaper type (£30-£40) mark are not that good and the pins on the tool have a tendency to bend if overtightened.
Wrap the chain round the sprockets and place the split link through to join it up. Don't forget to put the O rings in place if you have any, and use plenty of the grease normally supplied. You will then have to put the O rings and plate over the other side. You then place the chain/split link in the chain tool and squeeze the pins so the heads of the pins flatten out/dome over like the head of a rivet, to stop the plate falling off while riding down the road

This is where a lot of the cheaper tools fail. You need to get the pin on the tool square on and tighten it up so the end of the pins on the split link roll over. It needs a lot of pressure but not so much that the link does not move freely. I have found that the pins on the chain tool have a tendency to slip off as you tighten up, and the amount of pressure needed can bend the pins on the cheaper tools. So take it easy and go slowly but surly.
The last time I done a chain I managed to get the link pins bent but then the pin in the tool bent. Luckily enough it was OK for me to ride very carefully down the road to a motorcycle garage, where they very kindly (from a safety point of view) Finished off the last bit for me while I waited.
Hope that give you some insight
(:-})