Rule No. 1 for troubleshooting DVD player malfunctions is to clean the DVD itself first. It may just be that the disc you're using is scratched or smudged. Always try playing a few discs you can trust (ideally, you should test new ones) before monkeying with the DVD player.
If your DVD player is getting old and cranky (and not worth repairing), it might be time to open up the case and give the old insides a good dusting. Even if you've been diligently dusting the outer case and the vents for the past 10 years, there's still going to be buildup on the inside. If possible, try not to touch anything on the inside with your fingers, and don't drink soda from a leaky cup while you clean. But you knew that already.
Once you've got the case open, blast out dust from the corners with a compressed air canister. You can technically use a vacuum cleaner or a hair dryer to clean the inside of the case, but it might be safer not to do so. Both devices create a fair amount of static electricity, and while the risk is low when you're vacuuming or blowing away external dust, there's no reason to risk it. Compressed air canisters are cheap, static-free and easy to find at any electronics store.
We only advocate using this as a last resort, but you can actually clean the lens by hand. Again, don't touch it with your fingers, as the oils on your skin will smudge just about anything you touch. Instead, dip a cotton swab or a cotton ball in a tiny bit of rubbing alcohol to clean the lens, then dry it and pop in a working DVD to test it. Besides using just a little rubbing alcohol to clean the lens, you shouldn't use any solvents to clean your DVD player. The case of your DVD player will stain badly if you try to clean it with anything gentler than a mild soap.