To become a D-List celebrity, you need to gain fans. You do this by completing projects, gigs, and appearances. 5 stars will give you the most amount of fans gained for projects, so try to get 5 stars on your gigs! Dating also gives you fans but it is very costly. Just keep completing gigs, whether they be goal related or random ones that Simon calls you with until you reach the D-List. Causing some drama can also gain you fans, but be warned that you can also lose fans.
You will need around 5000 fans to reach the D-List.