Manfred, I will not pretend to have anything like as much video experience as you do - I'm just scratching the surface, but there are a number of tools available that can help overcome some DSLR drawbacks.
Magic Lantern: This is a fairly unobtrusive, free firmware hack available for a number of Canon cameras. It provides some video controls that current DSLRs lack. Manfred mentioned (among other things) focus peaking, over-exposure warning ("zebra stripes"), focus ramping, and microphone gain control, all of which are provided by Magic Lantern. The interface is not very intuitive, but with patience, it can help you patch some holes in DSLR video controls. It's available for the 60D, T3i, 5D, and a few others.
Canon STM lenses: Manfred also mentioned lenses with stepper motors to control focus. They're smooth, and essentially noiseless, so they're less likely to be picked up by nearby microphones (one of several reasons lots of video shooters use off-camera sound). All Canon lenses with an STM designation in their name have these motors. Currently, there are 3. The EF-S 18-135mm f3.5-5.6 IS STM, EF-S 18-55mm f3.5-4.5 IS STM, and EF 40mm f2.8 STM. Note that only the 40mm is compatible with full-frame cameras.
Shoe-mountable audio recorder: No current photo DSLR offers an XLR microphone input (they're huge compared with most camera auxiliary connections), which is the most common connection for professional mics. However, there are fairly inexpensive recorders which mount where a flash would normally go and can record dramatically cleaner sound than a camera's built-in microphone (which are all pretty poor). This one runs on AAs, has pretty darn good mics built in, provides phantom power, and can record from up to 4 mics simultaneously.
As you mentioned, your budget doesn't really allow for a dedicated video camera. Fortunately, in good light, the video quality from a T3i compares well to video from a 5D. Both kick out 1080p up to 30fps or 720p up to 60fps. Although, it's worth mentioning that the 5D will have considerably better low-light performance, and higher dynamic range. Many features that set high-end cameras apart are much less critical for video. You'll be focusing manually more often, so quick autofocus with lots of points isn't really necessary, and you don't need a high shutter drive speed.
If you're buying new, I would recommend looking at the following setup (1,070USD total).
- Canon T3i camera body with Magic Lantern (550USD or cheaper used)
- Canon 18-55mm f3.5-5.6 IS STM (250USD)
- Zoom H4n 4-track recorder (270USD)
I'm certainly not the world's greatest authority on video, but I think you'll have a hard time finding a better combination for the money. There are a number of accessories I'd recommend, but without knowing more about your needs, it's hard to say what will work best. Nevertheless, you will probably need the following(not necessarily immediately).
- Manual white balance tool (grey card or ExpoDisc)
- Video tripod with pan/tilt head. Recommend Manfrotto's mid and high-end 'pods, or a Gitzo.
- Microphones.
- "Dead kitten" (furry) wind noise cover for your audio recorder.
- Good editing software. Available pretty cheap if you happen to be a student.
- Stabilizer. There are dozens of rigs designed to keep SLRs stable while moving. I use a ghetto-fabbed GlideCam to film skiing, but what's right for you will depend on what you're shooting and how you're moving.
- Lots of camera batteries (2 minimum). Video runs the sensor continuously, so your camera won't last nearly as long shooting video as it would shooting photos.
Note that no current DSLR setup can change aperture smoothly, or zoom without borking focus. Those are two features of dedicated video cameras that the DSLR accessory market hasn't (to my knowledge) matched yet. But it's still possible to get really good quality with relatively inexpensive equipment. Manfred's points are all excellent, and there's no substitute for the right gear, but I think this should give you a very good starting point.