If I'm not mistaken, this is a rather well documented issue with that model. If memory serves, the heatsink on the GPU has a tendency to go on a bit of a walkabout. You're probably never going to fix this issue on your own.
I have a HP Pavillion DV6Z "entertainment laptop" that I ended up getting from my girlfriends sister for $50, she said she could never get it to work right, the screen wouldn't even come on. After a few updates, I eventually got the screen to turn on, however it still seems to have problems. Every so often when I turn it off for updates the computer's screen won't come up, however it will appear on an external monitor. And I have to hard reset it a few times for the screen to come up.
Sometimes it still goes black, especially when trying to play a game. For example: I start of KOTOR and I can make my character, then once the game actually starts, and it's playing a small cinematic (I click through the scroll) it may play for a second and then the screen is go black and the sound will stop. The computer will normally show up on an external monitor (when it goes black), but it won't when it goes black.
I have tried everything I know, and it doesn't seem to have helped, so I turn to you. I am willing to upgrade the computer (in fact I am about to upgrade the ram to 8gb and possibly the HDD to a terabyte).
Here's the specs.
Operating System: Windows 7 Home Premium 64-bit (6.1, Build 7601) Service Pack 1 (7601.win7sp1_gdr.120830-0333)
Language: English (Regional Setting: English)
System Manufacturer: Hewlett-Packard
System Model: HP Pavilion dv6 Notebook PC
BIOS: Default System BIOS
Processor: AMD Athlon(tm) II Dual-Core M320 (2 CPUs), ~2.1GHz
Memory: 3072MB RAM
Available OS Memory: 2812MB RAM
Page File: 1459MB used, 4162MB available
Card name: AMD M880G with ATI Mobility Radeon HD 4200
Manufacturer: ATI Technologies Inc.
Chip type: ATI display adapter (0x9712)
DAC type: Internal DAC(400MHz)
Device Key: Enum\PCI\VEN_1002&DEV_9712&SUBSYS_3635103C&REV_00
Display Memory: 1474 MB
Dedicated Memory: 324 MB
Shared Memory: 1150 MB
Current Mode: 1366 x 768 (32 bit) (60Hz)
Monitor Name: Generic PnP Monitor
Monitor Model: unknown
Monitor Id: AUO22EC
Native Mode: 1366 x 768(p) (60.042Hz)
Output Type: Internal
Driver Name: atiumd64.dll,atidxx64.dll,atiumdag,atidxx32,atiumdva,atiumd6a.cap,atitmm64.dll
Driver File Version: 8.14.0010.0678 (English)
Driver Version: 8.635.0.0
DDI Version: 10.1
Driver Model: WDDM 1.1