The D5000 shoots video at the following resolutions:
VGA 640x424 / 24 fps
QVGA 320x216 / 24fps
HD 1280x720 / 24 fps
VGA and QVGA are pretty much unusable these days - think of the crappy videos cell phones used to put out a few years ago. 1280x720 @ 24fps is fine.
The thing with shooting video on a DSLR is that you need decent lenses, probably more so than if you were just shooting stills. The ability to focus on objects further than 3 feet away is a function of the aperture you're using.
I don't know what lens you're currently using, but I've found that prime lenses work best with video from these DSLR's - even when compared to the top of the line zooms (L Series from Canon and Nikons pro range).
You can't go wrong with a good prime lens. The 50mm f1.8 is a good lens to start with, affordable, fast and very sharp. I'd get the D version and avoid the G. If you're finding it a bit long on your D5000 - as it's equivalent to a 75mm (50mm x 1.5 crop factor) then you can look into either a 24mm/28mm or a 35mm. But those will cost significantly more.
Hope that helps, let me know if you have any more questions!