Because of relativistic effects, from your point of view on the train you would just walk forward. But you would notice a strange effect while the trains were accelerating: your atomically synchronized wristwatch the clock you can see out the window has slowed down and stopped counting time. So it seems that your journey to the front of the train takes no time at all.
From someone standing on the side of the tracks catching a glimpse of you and the train as you whizz by, the front of the train is moving at light speed. You’re at the back of the train completely frozen still, unable to move forward because the front of the train is moving away at light speed.
Weird things happen when you’re talking about the limits of physical reality.
You are correct, I should have said there was an atomic clock out the window that the walker looked out at.