Please note: This Achievement can be Unlocked even if you do not own the NASCAR Expansion.Rolling Start races will occur naturally throughout the NASCAR-specific Career Mode, so if you are looking to complete the entire expansion then you don't need to focus on this Achievement in any great detail.
If you do want to specifically go for this Achievement however, the best way to do so is in Free Play, with the following Game Options. The track/car/lap selection is up to you of course, though I recommend a fast car (P or X-Class) and a short track in order to complete this in the fastest time possible.
Quick Stops (and the 2nd lap that is required for these) are optional but if you enable them you can get this Achievement at the same time:
Once you've completed a race you can select restart from the post-race menu and go again. Simply complete 5 races with these settings and you're done.
Combined, both Achievements should take around 10-15 minutes total (including loading times). I chose Road Atlanta Club and X-Class for my track/car combo.