What you'll need to do the job:
C wrench for single-sided swingarm bikes, socket wrench for double-sided or other adjusting tool provided in your tool kit.

If the tire doesn't roll freely once it's been cleaned, then the chain may have some tight spots -- a sign of age. Once you get too many it's time for a new chain. Or the chain may actually be adjusted too tightly. Generally, though, a chain stretches over time and becomes too loose -- sags and/or hits the swingarm excessively while the tire is spinning. Whether tight or loose, a chain can easily be adjusted to the correct tension.

To check tension consults your owner's manual for the tension millimeter listing. Sometimes it's listed on the swingarm itself. A chain should have about 1-1/2 inches of free play. To check, hold a ruler to the base of your swingarm and push on the chain. It shouldn't move with more than 1-1/2 inches of play.

For a double-sided swingarm, loosen the axle nuts of the rear wheel, which allow it to move for adjustment. On the end of each swingarm there is an adjustment bolt covered by a lock nut and perhaps a security cotter pin. Loosen the lock nuts on these bolts and then turn adjustments with the socket wrench equally on both sides. Since the axle nuts have been loosened the tire will move either forward or backward thus loosening or tightening the chain. It's important to match the markup lines on both sides since this assures that the wheel has been moved evenly on both sides. If you've reached the last markup line that means the chain is stretched out and it's time to replace.

For bikes with single-sided swingarm, use the "C-wrench" that should be part of the bike's tool kit. Turn the axle nut 1/4 to 1/6 of a turn to move the rear wheel back to tighten (or forward to loosen). It will move back click by click evenly.

Some bikes with a double-sided swingarm have their adjustment bolts on the side with a special tool provided in the tool kit to operate. The same procedure as described above applies