If multiple creatures enter under your control at the same time, Renegade Doppelganger's ability will trigger once for each of them. You choose the order that those abilities will resolve. It's possible to have Renegade Doppelganger become a copy of something, activate an activated ability of that creature, then have it become a copy of something else under this scenario. The last ability that resolves that you choose to use determines what it ends up a copy of.