Autodiagnostiqué is the past tense, first-person conjugation of the verb autodiagnosticar, meaning "to self-diagnose". It's an interesting and somewhat uncommon word that immediately grabs attention because it's a bit of a mouthful and not something you hear every day.
In the song, Camilo humorously uses this word to explain that since doctors can't find anything wrong with him, he's taken matters into his own hands. He "self-diagnosed" himself with a unique condition: being far from his loved one and missing them, creating a clever play on words with the song's title, "Autodiagnóstico".
Camilo turns lovesickness into a running medical joke. He opens with "Ya fui al doctor seis veces" [I've already been to the doctor six times], and every time the doctor tells him he's fine. So he self-diagnoses instead, naming a "bacteria muy rara" [very rare bacteria] called "tú estás lejos y te extraño" [you're far away and I miss you], and an illness he calls "no puedo olvidar tu caras" [I can't forget your faces]. It's a corny bit and he knows it, which is exactly why it works.
The punchline lands mid-song: "si me quitan esa enfermedad, me muero" [if they take that illness away from me, I die]. He'd rather stay sick than get cured, because the sickness is just love with a symptom list. Nobody in this song says "I love you" outright. Instead he begs, "Que no digas 'yo también'" [don't say 'me too'], because he wants to be the one who's caught, not the one who's matched.