Do you see an Error Code stated on the screen? If so state it here in a Comment.
I would like you to perform a simple step first, then we'll go on to diagnose.
I would like you to remove the Ram Memory module/s ('Stick'), clean the contacts, and then reinstall.
A) Remove ALL power from the laptop. Remove the AC adapter (Charger) and Battery.
B) Lay a towel, or similar object on your work table, and lay the laptop upside down on it.
(Bottom facing up)
C) De-Static yourself. Your body carries Static electricity. Static will fry out (Short Circuit) the hardware components inside a computer.
The best method is to wear an ESD wrist strap, and connect it's alligator clip to a good ground source. (Electro Static Discharge. An ESD wrist strap usually runs about $5)
Next best method is to place a large metallic object on your work table, and touch it Before working on your computer, and occasionally during.
(If you get up, and leave your computer repair in the middle, upon your return be SURE to touch the metallic object again. Metallic object? Perhaps a large metal serving tray, or metal knickknack)
D) Looking at the bottom of the laptop in the Middle, there is a large rectangular compartment panel.
The panel's length goes Vertically. There are two vents in this panel. One horizontally across the bottom, and one vertically towards the top.
This is the Memory Module Compartment
There are four Philips head screws to Loosen. One towards the top left corner. One below it, and two in the middle. (Left and Right)
Once the screws are loosened remove the Memory Module Compartment Panel.
(The cover swings up, from the back towards the front, of the laptop)
Gently spread the Retaining Tabs away from the Ram Memory module. Ease the Retaining Tab's 'bump' out of the notch on the memory module.
Once both of the 'bumps' are free from the ram memory module, the memory module will slightly pop-up.
The Ram Memory Module is NOT removed in a straight up position. Ease it out at an angle.
(Otherwise the ram memory module pins will be damaged, and the slot it goes in)
Use a pencil eraser, and clean each gold plated contact pin. Just a few strokes up, and down on each contact pin. (It may not look as though you have cleaned it very well, but I assure you that you have. Doesn't take much 'corrosion' to make a bad contact surface)
Use air to remove the eraser dust, reinstall the ram memory module/s.
Here is a free Service Manual from HP Support,
HP Support > Pavilion dv6-1030us Notebook PC > Manuals,http://h10025.www1.hp.com/ewfrf/wc/manualCategory?lc=en&dlc=en&cc=us&product=3860542&lang=en&
I would like you to scroll down, and click on the file name -
HP Pavilion dv6 Entertainment PC - Maintenance and Service Guide
This is a PDF file. It may take as much as 30 seconds before you will see the first page.
It may also take additional time to fully download.
Didn't fix the problem?
Use a flashlight, and shine it at the LCD screen with the laptop on. Shine at various angles. Do you
see faint images?
Laptop OFF, attach a VGA monitor (IF available) to the VGA port on the back of the laptop.
(CRT. Looks like a small TV. OR you can use a flat LCD screen monitor if it has a VGA cable)
Laptop ON, press and hold down on the Fn key, (Bottom left corner, second key towards the right, on the Keyboard), and tap the monitor toggle over key.
(Top left corner, fifth key from the left. Has a rectangular monitor icon on it)
Do you get graphics on the external monitor?
If the answer is yes to both procedures above, your graphics chip is working fine.
You have a problem with the Inverter.
An LCD screen cannot produce enough light by itself. It needs an additional light source.
A Backlight is the additional light source.
A Backlight is a CCFL.
Cold Cathode Fluorescent Lamp.
Similar to the Fluorescent lighting used in homes, and businesses, but on a MUCH smaller scale.
An Inverter is used to convert the electricity from the laptop, for the Backlight, and LCD screen.
90 percent of the time the fault is the Inverter.
The Inverter is located behind the Display Bezel, and at the bottom of the LCD screen. It is a Small rectangular circuit board.
(Page 84 and 86 in the Service Manual)
Very Easy to replace.
For additional questions please post in a Comment.