Orwyle was the Grand Maester during the final days of the reign of King Viserys I Targaryen. Upon the death of the King, Orwyle was present in the meeting in which Queen Alicent Hightower, Ser Otto Hightower - the Hand of the King - and Ser Criston Cole, the Lord Commander of the Kingsguard, forced the small council to crown Prince Aegon regardless of Princess Rhaenyra Targaryen being the designated heir. Orwyle witnessed the death of Lord Lyman Beesbury - who remained adamant in crowning Rhaenyra - at the hands of Ser Criston.
In the books
In the A Song of Ice and Fire novels there's no mention of Grand Maester Orwyle, who appears in The Princess and the Queen and The World of Ice and Fire. He succeeded Grand Maester Mellos in the year 127.
After Rhaenyra seized the Red Keep during the Dance of the Dragons, Orwyle was imprisoned in the black cells, where he wrote his account of the war, trying to put himself in the best light possible, uncertain if he would survive or not. In contrast to Eustace's pro-Aegon II writings, Orwyle's account was likely slanted in Rhaenyra's favor. Grand Maester Munkun based his The Dance of the Dragons, A True Telling in Orwyle's already biased account.
It is very probable that Orwyle was actually a secret Rhaenyra-loyalist since the beginning of the war. The print version of The Princess and the Queen had large sections cut out and doesn't contain the whole story of the Dance of the Dragons - and in the print version, Daemon Targaryen makes a passing comment that the Blacks secretly have a "friend" on the Green council itself. Of the six men on the Green council, Criston Cole was certainly Rhaenyra's enemy and died fighting her armies; Otto Hightower was executed when Kings' Landing fell (and as Alicent's father was certainly Rhaenyra's enemy); Ironrod was executed alongside Otto, and when to his death loudly insisting that a son inherits before a daughter (instead of pleading that he was a secret Rhaenyra loyalist); similarly, Tyland Lannister was handed over to the torturers, but refused to say where all the money from the old royal treasury was, much less admit that he was somehow a secret Rhaenyra loyalist. The only two remaining members were Lharys Clubfoot and Orwyle, but Clubfoot was Aegon II's most dangerous ally, secreting him and his children out of the capital when it fell, instead of just handing them over to the Blacks. Therefore, by process of elimination, the Blacks' "friend" on the Green council was almost certainly Orwyle - which does seem to match with how he was imprisoned instead of executed after King's Landing fell, and went so far as to write a pro-Rhaenyra account of the war while in captivity to try to ingratiate himself with her.