What would be the best way for city elves to fight for their rights?

They would have to bide their time, as a time of peace and prosperity would be the worst time to stage a coup. A time of Chaos would be the perfect time for a coup, because it's practically how Vladimir Lenin, Hitler, and Stalin came to power. Then they would have to gain enough sympathy from the upper class, so that their uprising would look justified. if they were to fight without these things, they would try and fail.

You've always got the problem how the rest of the population is going to respond. Take the war table mission for Lavellan. You can arrange for the assassination of the corrupt noble in charge of Wycome and his cronies subsequently flee, leaving the evidence of what was going on that convinces the local people to accept elves onto their ruling council. However, in the meantime the corrupt nobles have fled elsewhere in the Freemarches and persuaded them that it was all down to the rebellious elves and the evil Dalish and persuade them to take up arms against the ruling council in Wycome. Then it is only the timely intervention of the Inquisition forces that prevents a massacre. This is why you at least need to get the local commoners on side of all races if you are to have any chance of success. If you can get some noble minded aristocracy on side as well, so much the better. Otherwise the rest of the population will see the killing of the nobles as just an excuse to murder elves in retaliation.

The elven uprising in Halamshiral failed because it was confined to the elven population. The Chevaliers and nobles are known to abuse their power with the commoners generally, so if the leaders of the elven group had appealed to the wider peasant population they might have got somewhere. Then it would have been a general call for justice under the law and it would have been far harder for Justinia to urge Celene to putting it down without every commoner throughout Orlais feeling enraged or for Gaspard to capitalise on her initial apparent inaction with a satirical play using Andraste and Shartan. As it was, it was just seen as the Empress rightfully dealing with a bunch of rebellious elves, with all the old prejudices being used to justify it.