Let's start with the camera. Chances are that the phone in your pocket has a decent video camera built-in, especially if it's a new iPhone or Android. The main difference that you will see in moving up to a higher-end camera such as a dSLR is that the larger sensor will perform better in poor lighting conditions, and you will have greater control over depth-of-field in your shots. Depth of field (DOF) is the distance between the nearest and farthest objects in a scene that appear acceptably sharp in an image.

With a larger sensor, and the ability to choose zoom lenses, shooting with a DSLR allows you to do things in a head shot like keep your subject in focus, while creating a pleasantly blurry background, which helps isolate your subject and keep them the center of attention (see the attached videos for great examples of how this can be used to good effect).

If you decide that the advantages of a DSLR are worth stepping up for, there are a few routes to go, in order of preference. You can 1) borrow one from a generous friend for the shoot 2) Rent one 3) Buy one.

If you go the renting route, sites like Borrowlenses offer great deals on camera rentals; you could get the Nikon D7000, with a 50mm lens, for 3 days for $61 for the camera, and $20 for the lens; this is a great deal for a beginner setup. If you're looking to buy (and the D7000 is a fantastic all-around camera) you can grab one at Amazon here for less than $900, and a starter 50mm lens here for about $130 bucks.