dSLRs are worse for video than Sony or Panasonic mirrorless. Mirrorless are designed for things like live view (which is a lot like video), for focus from the main sensor (not a separate AF sensor), and so on. With dSLR, you're also paying for a mirrorbox and optical viewfinder, both of which are useless during video. Mirrorless have lenses designed to focus off of the main sensor. There are a lot of issues around ergonomics as well.
The first serious video from a still camera was a dSLR -- the Canon 5D MKII. It was the first still camera used professionally for things like TV shows. It got eclipsed for by the GH2 I suggested for all intents except DoF control. That's been used in a lot of pro video as well. That, in turn, recently got eclipsed by the Sony A7S/A7SII/A7RII, which are all very nice, but well outside your budget (Panasonic also released the GH3 and GH4, both of which are used professionally as well; I'm not very familiar with the GH3, but the GH4 is in a similar class to the Sonys, but also outside of your price range). Panasonic's technology has trickled down to lower-end models as well. Sony hasn't yet recently lower-end mirrorless with video in the same class as Panasonic, but that said, they're still quite good. Canon continues to update video features, but is a bit behind Panasonic/Sony. And Nikon/Pentax are a step down from there.
Although the truth of the matter is it might not matter. Everything on the market with audio level controls and a microphone input does embarrassingly well. Keep in mind that the Canon 5DMKII is now almost a decade old. Technology has come a long ways since then. You can see some of the rather spectacular shoots taken with it (Google around, or watch the finale of House which was filmed entirely with one). Video from most modern cameras will be even better. GH2 video is visibly sharper than the 5DMKII. Unlike the 5DMKII, it can autofocus during video shoots (which a pro would never do, but which is useful for home videos). It has better ergonomics. But at the end of the day, most cameras, again, with proper audio, are capable of producing spectacularly good video, whether dSLR, dSLT, or mirrorless.