At your price point, I'd have to recommend the Black Magic Pocket Cinema camera, though I don't believe it comes with a lens, so you would still need to buy a lens which would put you over your $1000 budget.
You could get a basic PDAF (phase detect auto focus) enabled video recording DSLR like a T4i along with a semi-decent zoom lens or a fairly good prime for $1000 as well, though the cheaper DSLRs (T4i is only one or two steps above entry level) tend to have pretty strong rolling shutter, which will produce a "jello" effect when panning.
I would recommend staying away from any DSLR that doesn't have PDAF enabled video for a beginner because it would mean having to manually adjust focus while shooting, which is fairly challenging to do well.
The Lumix GH3 is also pretty well regarded in terms of video quality for the price, though the GH3 is a bit older. I also don't recall for certain if it has the ability to auto-focus on video. It does produce very nice video quality for the price though.
One nice thing about shooting music videos is that the audio doesn't matter much. You can use the recorded version of the song for the sound and don't have to worry about sound from the video shoot, so you don't need to worry about a multi-track recorder (which would make your $1000 budget almost completely infeasible.)
You will still additionally probably want to get some lights though as good lighting is also pretty key to shooting music videos, though if you only have a $1000 budget, you are probably best to put it all in to the camera and use whatever lights you can find for now and then buy good lights later when you can.
Overall, $1000 is still very much on the cheap side for getting a rig capable of really professional quality video. Most people doing even fairly low end professional cinematography have multiple lenses that cost more than your entire camera budget, not to mention the cost of the actual camera.
