What can cause vibration and noise from the chain? I just replaced my front cogs and back sprockets with new ones (I didn't use a torque tool or replace the chain). I also replaced the sealed bearing bottom bracket on my pedals. The old bearings were 115 mm long and the new one is 117,5 mm long.

Now I feel some sort of vibration, and there is a noise like the gear wants to change, but the rear derailleur is set correctly with one click per gear. What can cause vibration and noise from the chain?
If you changed the crank length then the front derailer likely needs to be adjusted, and the rear may need a few tweaks.  Also, when you change the rear cluster there can be a difference of a mm or two between old cluster and new, requiring adjustment of the rear derailer.

Have you readjusted both derailers "by the book"?

(Whether or not you used a torque wrench would have little bearing on this issue, though you need to be sure to check that your crank bolts are tight every 200km or so for the next 1000km.)