If the system is under warranty, you should contact HP.
If it is not under warranty (which is probably the case, the DV6000 laptop series is rather old) many repair shops will charge several hundred dollars to even diagnose the system.
If there is a Fry's Electronics near you, I'd recommend them first. I've worked in their service departments, and they seem to hire technically proficient people. Best Buy and Geek Squad can be OK, but they tend to cost more and in my experience they do not hire people who are as technically proficient.
The possible points of failure that are causing this issue would be either Graphics, CPU, Mainboard, or Memory.
Beeping noises usually are signals sent by the laptop hardware, via the BIOS, rather than controlled by the operating system. You may want to do a Google search for "HP DV6000 beep codes" and see if you can find a code that corresponds to what you are hearing.
If the issue only occurs when you attempt to play a game it could be heat related. Try placing the laptop in the coolest room in your house, on a hard surface, with a fan blowing at it, and see if the beeping begins as quickly. If it does take longer to start beeping, the issue is probably cooling.
If you have more than one memory chip in the computer, try removing one and then the other and see if the beeping still occurs. This will indicate whether the issue is caused by your memory.
CPUs and mainboards and usually hard to come by and difficult to replace yourself, and so I do not recommend attempting to check these yourself.
Due to the system's age, though, it will very quickly become a question of whether or not it is worthwhile getting it repaired.
If the laptop is older than 3 years and the quoted cost of diagnosis and repair is more than 1/3 to 1/2 of the original cost of the laptop, it will probably not be a good idea to try and repair it. Of course, the specific point at which it is no longer worthwhile to repair is unique to you and should be decided by you.