A stiff link doesn't necessarily have to be removed. A good shot of WD40 on the offending link will help to remove any dirt or grit that may be causing the problem. Working the rivet out 2 or 3 turns then back again might solve it.
If you replace a link on an old chain with a new link you may get the same problem. And if you replace the chain then you'll still probably end up with a chain that jumps as you pedal.
If you replace the chain you will also need to replace the rear gear sprockets.
This is because the chain and the sprocket wear down in unison, meshing into each other. A new chain won't mesh. A new link won't mesh. Even using a link from an old chain won't work.
To replace a chain you need a chain break tool.
Unscrew the bolt in the tool and set the chain into the teeth on the tool. Screw the bolt in until it touches the rivet you wish to remove. Turn the bolt no more than 6 full revolutions. The rivet should still be attached to the outer link.
If you turn more than 6 revolutions you will push the rivet out of the outer link and will never be able to put it back in place!!
Open a link on one end of the new chain before sizing it beside the old chain. (Count the links as the old chain will have stretched!)
To replace a rivet, screw the bolt on the tool nearly all the way out to leave room for the projecting rivet, reverse the tool, set the links into the teeth on the tool, oil the joint, and screw the bolt inwards until the rivet is in place and the chain is supple.
If the new joint is stiff, unscrew the rivet about a revolution and back again.
Make sure the rivet projects equally from each side of the link.
Big Softy, November 2006