To detect the keylogger problem, try changing the password from another (possibly safer) computer, if you're on Windows, try Linux or a Mac. Then don't login to the account using the "unsafe" computer. If the problem persists we can conclude the problem is not tied to a specific machine and it's probably not a keylogger on that system.
Another option could be that the "reset your password" emails are forwarded to another account which someone else can read.
- Check for any filters with forwards and delete them
- Remove any forwards (there is one forward option in Gmail without filter).
- Change the passwords (again on a safe computer) for the alternate email account.
Gmail logs from which ip-address your Gmail is visited (look at the footer).
"Last account activity: 17 minutes ago on this computer. Details" check that for unknown ip-addresses.