Welcome to the site. I believe your best chance for an upgrade is as a cash customer (vs. points customer). I have noticed that Marriott is migrating towards charging cash for upgrades to point stays (especially in the UK and Western Europe). This doesn't mean you can't score a complimentary upgrade, but they seem to becoming 'far'er and fewer between, according to folks here. Take a look at some of the discussion threads to the right under the heading 'More Like This' for further insight. There are other variables too, which may affect the outcome. Namely check-in time of day, day of week, peak vs. non-peak travel season, and most importantly I think, duration of stay. The consensus is that shorter stays are more likely to receive an upgrade. Having said all of that, there are many types of upgrades: high floor, view, suite(s) of all kinds, CL level, and in Europe sometimes two beds (vs. one) are considered an upgrade. Don't know if this helps.
I have to admit that I have actually had more success getting a room upgrade when using points. I had not thought of it before huey123 asked the question. Possibly the reason is that most of my point stays are for personal/vacation travel and I take the time to contact the hotel in advance. I have never actually asked for an upgrade, only a "nice room." Most times I have received a really nice room - an upgrade I am sure.
I have been told the best way is to ask. And I assume either call ahead to ask or arrive as early as possible for check in. Good luck and please post if you have success. I've only received an upgrade once and travel almost twice a month. It's disappointing to know it's available and not offered.
I'm going on my honeymoon in November and e-mailed a couple of the properties at the time of reservation and cordially asked for an upgrade. I mentioned that I am a Plat and the circumstances for travel, but also included a mild degree of flattery and a great deal of gratitude. It worked and I have received confirmed upgrades to larger rooms, though not suites. That's fine, though, as I was primarily concerned with being stuck in a room the size of a cruise ship cabin. I've read that this works under other circumstances or no circumstances at all, but chance are better if you do have a special circumstance and it's a property not in the US. For some reason I tried for my wedding of all things and they wanted to charge me $400 more (JW Marriott). On the other hand, I've stayed at the Waikiki Beach Marriott and received complimentary upgrades to better views and one time a Jr. Suite numerous times. If you do go here, take the view upgrade. The Jr. Suites are nothing special.
It's worth noting that my Waikiki Beach Marriott stays were paid and certificate, while the honeymoon stays are all points.