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.