How did they go from being a permanent member in the Security Council to not being a member?
Taiwan wasn't a member in the sense that the UN decided to give one little island a permanent seat. They were recognized as the true government of all of China, and had a strong presence there until they were eventually forced to flee the mainland to Taiwan in 1949. By 1971 the People's Republic (communist China) was obviously there for good, and gained more international recognition, so China's seat was essentially given to them instead of the government in Taiwan.
One key factor would be its oppression towards the PRC. Because ROC claimed the whole of China, it could not allow PRC to join the UN as a separate nation, nor could it recognize the PRC's sovereignty.
Since admission was impossible, the PRC aimed to replace the ROC.
They went about this both in international diplomacy (making other countries switch recognition from ROC to PRC), as well as in introducing new nations into the UN to dilute ROC camp's vote. The motion to replace ROC was brought up every year by members friendly to the PRC. Finally in 1971, resolution 2578 was passed at 76 vs 35, and the UN itself switched recognition from the ROC to the PRC.
Since it is a switch in recognition, instead of admission of a new nation, the PRC replaced ROC's membership and hence its seat on the Security Council.
If the ROC had recognized the PRC and allowed it to join the UN as a member state, it's possible that Taiwan would still be a member of the Security Council, or at least an UN member.