In order for your Nikon to auto focus correctly, you need to have several things:
First - focus indicator. In your viewfinder, locate it on the bottom row of the status information bar; you should see a small, green circle in the left-most position. This will BLINK when you partially depress the shutter release (or back button focus - depending how you focus your D5000) whenever the subject is OUT of focus. It will switch to "STEADY ON" when the subject is IN focus. Please note that you will not be able to release the shutter any time it is blinking unless you make a change to the setting that will allow you to release the shutter when the camera is out of focus (not recommended).
Second - light. You must have enough light entering the camera for it to detect areas of different contrast on the selected focus point. If there is not enough light there is too little contrast for the camera to autofocus and no matter how close or how far (and everywhere between) the focus is set resulting in the camera "hunting" near and far to find it. The green circle will not stop blinking. Have you tried in a well lit room or outdoors in daylight?
Third - minimum focus distance. All lenses have a minimum focus distance. Acording to Nikon, yours is rated at 3.6' (1.1m) (see: http://www.nikonusa.com/en/Nikon-Products/Product/Camera-Lenses/AF-S-DX-VR-Zoom-Nikkor-55-200mm-f%252F4-5.6G-IF-ED.html
for full specs on this lens). This means your subject needs to be at least 3.6' away to be able to focus on it - manually or aut0matically - it makes no difference. There are lenses that do "very close" focussing that are called "Macro" (but curiously Nikon branded versions of these lenses are called "Micro"), that are also telephoto lenses. Your lens is not a macro/micro type.
Lastly - operational equipment. It is possible that there is an issue with the D5000 camera body or lens (or even both). Try another lens to see it it works as expected. No other lens? Go to camera store with your camera and lens and "shop" for a new lens. Ask to try it out in the store and see if it it works. Maybe the counterperson will identify another issue with your set up.
I hope this helps & good luck!