Nikon d5000 screen won't turn on

Have you tries cleaning the battery and camera terminals with some isopropyl alcohol and a qtip?

The D5000 does not have a focus motor in the body, relying on the lens to have one. The old 70-300mm G lens does not have a focus motor, relying on the camera to have one. Thus, this combination will not autofocus.
The newer 70-300mm AF-S G lens, as well as the even newer 70-300mm VR AF-S lens, do have the autofocus motor and thus will autofocus on the D5000.

Follow the guide below on how to replace the LCD screen on a Nikon D5000
(do not delete)

OR

Your camera is still under the one year Nikon warranty. Call Nikon's repair depot at 1-800-NIKON-US (1-800-645-6687) 9AM-8PM EST, Monday to Friday. They will verify your warranty and assist you in returning it for repair.

Just found this on the Nikon support website:-

My D5000 does not seem to respond to controls. Can I reset the camera?

In extremely rare instances, unusual characters may appear in the monitor and the camera may stop functioning. In most cases, this phenomenon is caused by a strong external static charge. Turn the camera off, remove and reinsert the battery, taking care to avoid burns, and turn the camera on again, or if you are using an AC adapter (available separately), disconnect and reconnect the adapter and turn the camera on again.
If the problem persists, turn the camera off, open the memory card slot cover, and using a paper clip, gently press the reset button by the memory card slot. ( NOTE: This will also reset the camera clock. Also note that disconnecting the power source or pressing the reset button may result in loss of any data not recorded to the memory card at the time the problem occurred. Data already recorded to the card will not be affected. In the event of continued malfunction, contact your retailer or Nikon-authorized service representative.

There's more info here:- http://support.nikonusa.com/app/answers/list/p/19

Just type D5000 in the search box

You should still be able to see the aperture through the regular viewfinder. When you say 'blue', you mean the whole thing is just blue? No writing? If that is the case, try a fully charged battery, full reset and if no change, I hope it's still under warranty. Get it to a Nikon service point.

The Nikon D5000 must be set in "Live View" mode before you can use the rear monitor to take photos. There is a button that says "LV" to the right of the LCD screen that sets Live View to on or off.

Are you in the green "auto" mode? If so, are your images underexposed when viewed on a computer screen?

Hey check out this video. the solution to the d5000 is same. the hook is a bit different though on the right.

This is a firmware problem that has been happening to so many people's digital cameras. The company will repair it for $50