If anyone can tell me if this is normal, I'll deal. If not,
Two things I saw. In the second when you were walking towards him the gun should not have moved like that.
In the earlier part you were drifting way off to the right and I don't know if that was you or not. You could try this. Change the batteries in your controls (or keyboard mouse if that is what you have).
Also, you might try fiddling with the Sensitivity settings, I find this game is very twitchy when I use high sensitivity settings.
However, that second part is not normal. I have only noticed gun sway like that when trying to snipe from far away and not to that magnitude. I see some gun sway close up, but not enough for me to miss at the distance you were from the enemy.