Let's say Andronikos I is overthrown by Isaac Angelos but manages to flee Constantinople.
While this is happening (mid 1185), the Norman invasion of the Eastern Roman Empire is in full swing and Thessaloniki, second city of the Empire, was just conquered and horrifyingly sacked (bonus: the incompetent defender of Thessaloniki was also a Komnenos).
So somehow Andronikos links up with the Normans and promises them the XII century equivalent of a billion dollars to help him take back the throne. Helped by infighting and instability in the Empire, the Normans and Andronikos enter Constantinople in 1186. The increasingly paranoid Emperor Andronikos introduces a reign of terror which makes his earlier exploits look positively benevolent, setting out to brutally murder the aristocracy in general and the Angeloi in particular. But the Normans themselves are getting less reliable and more hostile every day as Andronikos finds out he doesn't actually have the money he needs. At some point he attempts to drive the Normans out with force, but it backfires, and we get an early Sack of Constantinople and temporary destruction of the Byzantine Empire in late 1186 or early 1187.
After all that, Andronikos I Komnenos is reviled as a callous bastard and a madman. He's remembered chiefly as the man who joined the Westerners against his own realm, won the throne with their aid, and then provoked them into the Sack of Constantinople, slaughtering thousands and leaving the empire scarred for centuries. His struggle against the power of the nobility is referred to universally in terms of "tyranny", "brutality", "repression", and as a bonus he's probably blamed for the Bulgarian revolt too.
Meanwhile, Isaac II Angelos has ruled for a grand total of one year, and thus never had the time to show off his worst qualities. He is considered a pretty cool guy who stood up to the tyranny of Andronikos, tried to defend the Empire against the western menace and unfortunately died in this unfair fight. The Angeloi were decent folk and innocent martyrs of the last months of Andronik's reign of terror.
...seriously though this is a tough AHC. Even in the scenario outlined above (which is by no means perfect), only the last of the Komnenoi gets a ruined reputation. It will taint the dynasty as a whole, but not enough to make everyone forget their earlier achievements.