Two possibilities exist for issues like this. Either it's a display panel issue or it's a GPU/IGP issue. The former can be tested for by plugging in an external monitor and checking the results. If you get a normal image on the external display, your display panel needs to be replaced. If you get the same garbled display on the external, you have a GPU/IGP issue and need to replace the motherboard and/or CPU depending on the specific config of the unit.
Given the relative costs of both repairs and the fact that HP has turned Compaq into their low budget brand, both repairs are likely going to cost more than 75% of a brand new unit.
So the first thing I'd try is seeing if you can convince someone to give you a break on the warranty if it's within a month or so of the 1-year mark. It's a little advertised fact, but pretty much every manufacturer really has a 1.5 year warranty on products. They give retailers a 6-month "shelf warranty" for when the unit is sitting on a shelf somewhere waiting to be sold. So in a lot of cases, companies will give you a grace period on the warranty because of this. They don't tell you that it's SOP, rather they let you believe that they're making some kind of special exception just for you. If they won't budge on the warranty, I'd just write the thing off. Sell it on eBay to help defray the costs of a new unit a bit more and maybe this time don't buy one of the cheapest laptops you can find on the market. If it were me, I wouldn't even buy anything under HP's name brand, let alone what they've turned the Compaq brand into after acquiring it about a decade ago. I'd also avoid all companies that do limited run schemes, so no Lenovo, Acer (which includes Gateway, eMachines and Packard Bell among others) and ASUS. Getting repair parts for those systems after about 6 months is tricky and the older a unit is, the less likely it is you'll ever get them to honor a warranty. If you ask me, Dell and Toshiba are about the only companies left worth considering for laptops anymore. Companies like Sony, Apple and Fujitsu are just ridiculously overpriced for what you get and Apple has been actively making their systems less and less DIY repair friendly trying to get you to buy their AppleCare warranty and bring the unit to their Apple stores for repair through what I would consider unfair business practices. Samsung is still kind of an unknown because they're still pretty new to the computer business.
If you take absolutely nothing else away from this it's to never buy a computer with price as your primary concern. Laptops in particular will burn you in a big way if you walk into some store and just buy the cheapest thing you can find, which is what people do if they're buying a Compaq brand. HP only tends to sell those through office supply stores like Office Depot and Office Max. Even Best Buy seems to have stopped bothering to carry them and I'm not even sure if HP sells them online via its own site.