Try disabling the tTrackpad (you may need a working mouse as you will lose control without the trackpad!) Usually on Laptops this can be done using an F (then you will not need a mouse!!) If the problem does not occur when typing, turn the Trackpad on and see what happens.
At least this will indicate if the problem is with the Trackpad or not.
If it does appear to be the Trackpad...
Try cleaning the Trackpad, but do not use any cleaner that can leave a deposit on it (ie furniture polish). If using a liquid cleaner slightly dampen a cloth with it (that is do not our liquid on the Trackpad).
Thouroughly dry with a clean dry cloth.
Re Drivers.
I have on occasion found removing the updated drivers and reinstalling the ones that came with the computer has fixed a problem. Usually HP put this in a folder on the C drive.
Also only update using Drivers from the HP site, not generic drivers.
Hope that helps