I used to think that Esav killed Nimrod and took the special garment that was originally Adam Harishon's. (Is this a Midrash?) But now I'm wondering if Avraham Avinu and Eliezer killed Nimrod during the battle against the four kings? Thank you
The Pirkei D'Rabbi Eliezer does indeed state that Esav killed Nimrod (ed. Higger ch. 31). This is also found in Yalkut Shimoni (Parashat Tol'dot 110) and Targum Pseudo-Jonathan (Parashat Tol'dot 25:27). It is further quoted by Rabbenu Hayyim Paltiel (Parashat Tol'dot 25:32) and Rabbenu Bahya (B'reshit 3:21)
You are also correct that there is a Midrash that Avraham killed Nimrod. This can be found in Tanhuma (ed. Buber) Parashat Vayera (2), which identifies Nimrod with Amrafel, and states that Avraham killed him. This is also found in Yalkut Shimoni Yeshaya (511).