What is causing Kirito's eyes to change colour? In Sword Art Online, there are a number of times Kirito's eye colour changes. Originally in SAO and ALO, his eyes were kinda a grey colour:





When Kirito is killed by Heathcliff though, his eyes go golden when he comes back to life:



Now, I could just assume the eye colour change was a sign of him surpassing the system with his will (as acknowledged by Kayaba).

However, in episode 22 of season 1, when his HP is going down after being impaled by a Guardian's sword, as he is struggling to reach the door inside the World Tree, before he dies, his eye colour has changed. It also changes again during the Raid Battle in the World Tree when he produces that attack which allows him to breach the wall of Guardians. In these two instances, I don't see how his will is surpassing.

Considering also that his Avatar Data at this time was the SAO Data which was made to look like him, what is causing Kirito's eyes to change colour?
From what I have gathered, I couldn't find any information of why his eye color changed, but from what I can see, the most likely reason is as you said, his will was strong enough to surpass the system. The reason probably is of course Asuna.

From the three cases that you have mentioned in your question, Asuna was involved in all of them:


When he was killed, Asuna was killed right before he was, he got too angry and surpassed the system.
The second case where he battled but failed is because Asuna is so close, he is so desperate to see her again and he surpassed the system (maybe even with some help from Yui).
It was also the same with the third case, except this time with more help, he was able to defeat the guardians and reach the World Tree.