Although it's presumably out of warranty, I would take it back to the dealer and ask what they intend to do about it. A computer is a consumer durable and you are entitled to expect it to continue function for years after any warranty period.
Did you take it to the shop?-- because your contract is with the vendor, not the manufacturer.
Get that issue straight before you proceed. The warranty is from HP but that has nothing to do with your contract with the vendor unless HP sold it direct to you -- when they are the vendor.
I would write to the vendor saying you are prepared to take them to court for the balance of the value of the machine -- the way to work it out would be to calculate the depreciation over 6 years and if (for example) you've had it 2 years claim 66% of the purchase price.
If your jurisdiction has a small-claims-court system this limits costs so that the full court costs will not be met by you, even if you lose.