Undoubtedly the most widespread and critically acclaimed serial killer movie is
The Silence of the Lambs. The acting, direction, dialogue, cinematography, and nature of the characters all coalesce into a fantastically terrifying portrayal of madness (both beautiful - Hannibal, and perverse - Bill). Unfortunately I think the films that followed were rather haphazard and spoil a great character.
Manhunter, which was the first filmic adaptation of Harris' Hannibal, wasn't the most profound movie but nonetheless established the character as one of the nastiest antagonists out there.
The Silence of the Lambs followed and allowed for the predictable sequels.
Hannibal had its moments of gory goofiness but ultimately wasn't anything special.
Red Dragon moved the franchise into a different direction and was interesting, but something seemed to be missing. Lastly
Hannibal Rising, a prequel following the birth of the man, is intriguing in its execution and beautifully filmed yet lacks that certain something that made
The Silence of the Lambs so incredible.