The first Emperor (Isaac II) was couped by his brother (Alexios III who styled himself Alexios III Komnenoi even though he was part of the Angeloi dynasty) in 1195 after about 10 years on the throne due to the rising unpopularity of Isaac II after multiple botched campaigns in the Balkans. Alexios III proclaimed himself emperor while Isaac II was out of Constantinople on a hunting trip, had his brother captured, blinded him, and then imprisoned him. Alexios II son, Alexios IV, was not touched but was smuggled out of the city in 1201 by Pisan sailors who took him to Pisa, where he then agitated and sought support for anyone who would be willing to help him reclaim his family's throne from his uncle. He was able to wind up convincing Boniface of Montferrat (who was brother of Renier Montferrat, who had been married to Maria Prophyrogenita, daughter of Manouel I, and had been killed by Andronikos I alongside his wife during his purge of the Manouel branch of the Komnenoi) to support his claim just as Boniface became an influential leader (because the prior leader, Count Thibaut of Champagne, just randomly, suddenly dropped dead at a young age) in a new crusade to take Egypt so that it could serve as a springbed for a following, future crusade to take the Holy Land again - this was the Fourth Crusade. They managed to conspire to hijack this crusade - which was its own huge mess, with the vast majority of the soldiery not even knowing that they were going to Egypt instead of the Holy Land, let alone that their leaders had hijacked this already convoluted crusade and that they were instead about to be redirected to siege, overthrow, and then loot the most venerable and splendorous city in the world - to siege Constantinople, where they easily landed, Alexios III fled the city on news that a new crusading army had arrived to install his nephew Alexios IV. Alexios II was freed and made emperor again, but the Crusaders demanded that Alexios IV also be elevated to co-emperor, which he was.
But the two Angeloi were unable to pay the Crusaders their promised price and after a bunch of shenanigans which included the Crusaders staying around the city for a lot longer than initially planned and the Angeloi increasingly unpopular with the denizens of the city (in large part because their reliance on the crusaders to maintain order and their power was seen as surrendering power to the Latins, who were already unpopular in the city due to a bunch of other historical stuff that had been building for like over a century at this point, and it didn't help that they were actively sacking Thrace every so often to help offset their costs) while Isaac II was actively spreading delegitimizing rumors about his own son because he was upset that he had to share his throne with his son and that his son was wielding more power and influence than him. This then led to a popular revolt where an obscure nobleman, Nicholas Kanabos, was acclaimed emperor by the crowd, but refused the crown. Meanwhile the rebellion led to the Angeloi barricading themselves in their palace and sending Alexios V Doukas, aka Alexios Doukas Murtzuphlus (not an Angeloi . . . yet), to go get the Crusaders to go capture and murder Kanabos and put down the rebellion, and instead he bribed the Varangians guarding the palace to arrest the emperors, and he then proceeded to have Alexios IV strangled while in prison and Isaac II died around the same time (likely murdered also, either by poison or some other means, though the actual cause is unknown and is sometimes attributed to his frail body failing due to the shock at the coup/his son's death also). Kanabos was then allegedly offered an office under by Alexios V, which he refused, then refused the multiple summons by the emperor, fled to the Hagia Sophia, and was dragged out and murdered at its steps. Alexios V then led the failed attempt to hold back the Crusaders, who would capture and loot Constantinople and carve up Byzantium into fiefs, creating what is now known as the "Latin Empire".
Alexios V, accompanied by Eudokia Angelina and her mother - daughter and wife of Alexios III, respectively - then fled on a fishing boat to a Thracian city where Alexios III had holed up with some troops and asked to be let in, where he was welcomed and married off to Alexios III's daughter, Eudokia Angelina - who he had helped escape alongside himself - and thus making him part of the Angeloi dynasty by marriage. But Alexios III would later betray Alexios V by having him captured, blinded and leaving him for the Crusaders as he - Alexios III - fled Thrace for Thessaly due to a Crusader advance against Byzantine Greece - the blind Alexios V was captured by the Latins who then gave him a mock trial for the crime of treason and killing his liege (Alexios IV), found him guilty, and executed him by throwing him off of the Column of Theodosius. Alexios III was eventually captured and imprisoned by the Latins (by Boniface specifically) and eventually ransomed off to his son in law Theodore I Laskaris, the Emperor of Nicaea (the Byzantine successor rump state in Western Anatolia). He then started conspiring against his own son in law because he was upset that he wasn't still emperor and wanted to take power over Laskaris and launched a rebellion against his own son in law where he even teamed up with the Sultan of Rum to this end, but Laskaris managed to miraculously beat them in battle (even more miraculously also managing to kill the Sultan in the process, causing them to have their own issues and buying him some room to breathe), and Alexios III was finally arrested and confined to a monastery until he died later that year.
Theodore Laskaris is also a funny guy. He was brother to Constantine Laskaris who MAY have been crowned emperor. During the sacking of Constantinople, after Alexios V had fled, a bunch of people and Varangian guard gathered in the Hagia Sophia, the two brothers nominated themselves, neither figure could get the acclamation necessary, and so lots were cast by the troops and Constantine wound up winning out. But also that may not have happened, and he may just have been offered the crown and refused it. Anyways, he allegedly tried to inspire a final stand in the Varangian guard who refused to do so. He then succeeded in fleeing the city with his brother Theodore in tow. They escaped to Nicaea where he secured the Anatolian holdings. But there is some doubt to this and it may be that Theodore was actually the one crowned.
ANYWAYS, the brothers fled to Bithynia and set up the Empire of Nicaea, a Byzantine rump state and one of three Byzantine successor rump states (alongside the Despotate of Epirus - founded by Komnenos Doukas branch of the Angeloi, and the Empire of Trebizond - founded by grandson of arch-kin-slaying-guy Andronikos I Komnenos, Alexios I Megas Komnenos - where coups and familial fighting and assassinations were endemic after the first several emperors, but especially after the 9th). It eventually managed to succeed in reclaiming Byzantium from the Latins, but at the last minute the Laskarids were basically couped by Michael Palailogos who blinded and imprisoned the child emperor (like 7 or 8 year old) John IV Doukas Laskaris and enthroned his own family as the Palaologos. The Palailogos would then go on to have nearly 2 centuries of their own fractious rule with lots of familial infighting and violence that resulted in the final end of the Byzantine Empire.
