SEPTEMBER 2010 UPDATE: Both Gotham TV and the Tournaments section produce errors when you go to them now, with no announcement by Microsoft, and this site has listed them as discontinued. In case this is only temporary, however, I will leave the old solution in place.
I thought I'd share my experience getting this somewhat elusive achievement since as of mid 2009 it is almost impossible to randomly go on to Gotham TV and actually find feeds to watch.
First I'll say that that I have read that you don't need to actually save a replay of a race for this to unlock (despite a couple guides saying that you do), you only have to watch Gotham TV 10 times. When I unlocked this I had already saved a race, so I can't confirm if this is true or not. What I can confirm, however, is watching your own saved replay 10 times will not work to unlock this achievement.
The best way to unlock this in my experience is to go into "Gotham Career" and then go to "Tournaments". There should be several tournaments listed. Go to a tournament that has a status of "Finished Qualifying" or "Elimination". If there are none in progress, you will have to go to one with a status of "Qualifying". If the only tournaments available have a "Qualifying" status, note the qualification ending date and time and come on shortly after the listed date and time. The status should now be "Finished Qualifying". Note that the time PGR lists is 7 hours ahead of Pacific (US) time, so make sure to account for that.
At this point, I'm going to assume you have found a tournament that has a "Finished Qualifying" or "Eimination" status. Now, it will give you a message saying you didn't qualify for the tournament so you can't participate. Press OK and look around at the bracket and find the soonest (or whatever time will work for you) race time and note down the day and time and log on to PGR3 at that time. It will say what time it is for you according to the PGR clock on the top right in case you need some help figuring out time differences. For me, There was an event scheduled for 02:00 (7 PM Pacific) so I logged on at that time and I kept refreshing the tournament channel. About 15 minutes later it finally found a feed, and the feed only stayed up for about 20 minutes, so you are going to have a short window in which to accomplish this. I started up the event and abandoned the event 10 times, exited to dashboard, started PGR3 back up, and didn't get the achievement. I had to actually turn my 360 off and turn it back on again before it gave me the achievement, so try this if simply exiting to dashboard doesn't work. One final thing I will note is that the event I watched was the tournament final, so I can't confirm if any or all of the quarterfinal matches or semifinal matches will actually appear on the Gotham TV tournament channel. If not, you will have to log on at the date and time of the final, but hopefully it will work for the earlier events. Good luck!