A Temporary Resident Permit is basically a visitor's visa. They are saying that you can't take advantage any longer of the unique relationship enjoyed by Canadians and Americans in being able to cross the border virtually at will without having specific paperwork. You are now required to have the paperwork. You should be able to apply for a Temporary Resident Visa at the Canadian Consulate in Detroit (in the Renaissance Center). You can also download the application form here:
http://www.cic.gc.ca/english/visit/apply-how.aspGetting married would complicate things immensely and would not change the requirement for a Temporary Resident permit. You can certainly marry if you and your girlfriend wish to do so, but you and your wife would then have to start joint paperwork to obtain the necessary 'permanent resident' status that would allow you to live in Canada. You 'may' be able to apply from within Canada after you have received a temporary resident permit, but if you left Canada while your PR status is being processed (ie to work, etc.) you would probably be denied entry until your PR status was approved. The DUI convictions complicate the process. Ideally, with the DUI convictions you would be best advised to apply for the out-of-country processing of the Permanent Resident card. There may be requirements to prove 'rehabilitation' and there may also be a time limit that must pass without a re-offence before you are allowed to apply (usually 5 years).
So, check out the site, review the information, then apply for the Permanent Resident permit from the Canadian Consulate to be allowed to visit your girlfriend in Canada. Good luck.