After the events of Chamber of Secrets, it was clear to Dumbledore that Lucius Malfoy was behind opening the Chamber of Secrets (Dumbledore believed Harry & it seemed all the time that he kept an eye on Harry too). In subsequent years, he also discovered that Tom Riddle's diary was one of Voldemort's horcruxesHorcruxes.
Why didn't Dumbledore suspect that Lucius Malfoy was a Death Eater? We didn't see him confront Lucius or at least, raise his voice against Lucius in the Ministry.