Kool Moe Dee's beef with LL Cool J was because Moe Dee said that Cool J stole his aggressive stance and rapping style. On the cover of his 1987 album, How Ya Like Me Now, he had a Kangol hat (LL Cool J's trademark) being crushed by a Jeep. Kool Moe Dee was a harsh critic of other MCs. He used to issue "report cards" in the liner notes of his albums. When grading the Beastie Boys, he gave them mostly C's, but gave them an A for "sticking to themes".

Kool Moe Dee was born Mohandas Dewese in 1963. He had success early in hip-hop with The Treacherous 3. After this group broke up, he enrolled in college and earned a commmunications degree from SUNY. In 1989 he became the first rapper at the Grammy Awards ceremonies.

sources: allmusic.com and other sites on the net