I recently replaced the display in one of my laptops (a HP nx6125). I'm getting alternate-line vertical banding, it's almost unreadable.

Does anyone have any exprience with this?
It's not weird, it's broken - it could be the display, the cable to the display or possibly even the output from the GPU's DAC - either way it needs fixing, contact your service provider.
This could be happening in these conditions:
Have you tried to connect your laptop to an external monitor to see if it displays correctly? If so, that means the LCD is defective, if not, your video card is dying.
Did you replace the previous display screen for the same issue? if so, then you have a problem with the connector, ribbon cable, etc. if not, then double check the connections and try again. if the problem persists get a new display screen.
I think it's most likely the inverter that's about to break. It seems like the inverter fails for every second pixel column on your screen.
All LCD screens must rotate the liquid crystals in alternating directions for each refresh so that the crystals don't get "stuck". Different panels use different schemes for rotating neighboring pixels or lines in opposite directions to minimize the unfortunate artifacts it creates (rendering some cheaper displays completely unable to display alternating pixel patterns, particularly if they use overdrive to turn crystals faster).