First, changing MX records wouldn't be done in Gmail, it would have to be done through your DNS provider (the service hosting your domain's zone file). That's most likely the place you registered your domain name but not necessarily, that depends on how you decided to set things up.
Second, even if you change the MX records of your domain and point them to Google's servers, Google wouldn't know what to do with those emails when it gets them. That's because they would be addressed to an email address they don't know about. The only way to have Google handle emails addressed using your own domain is to set up a Google Apps account with that domain.
If you don't want to pay for the longer free Google Apps service, the only workaround is to do what the other answer explains: 1) use an intermediate service to accepts incoming emails addressed to your domain and set it up to forward everything to your Gmail address, 2) set up your Gmail account to be able to send emails on behalf of another address