If there are 4 people playing cards and each round consists of each one of them putting one card on the table, then once there are 4 cards on the table then that was a "go round". Then once there are 8 cards on the table, that was another "go round".
It's like a carousel (fairground ride). One entire circuit would be a "go round".
Another example: a group of 8 students are sitting in a circle. The teacher says that they will now go round the group and each student has to say the name of a flower. The teacher points at one student who becomes number 1. They go round the group until each one of them has said the name of a flower. On the next "go round" the teachers tells them all to say a boy's name. After 8 names, the teacher says "OK, on the next go round, I want you all to say the name of an animal."
In your excerpt, it basically means that it was his turn to die. Without knowing the rest of the book, I don't know if other people have died in the book and it's saying that this "go round" Tom dies. Perhaps on the last "go round" someone else died.
Student or Learner