Run your mic through a guitar amp with overdrive and increase the pregain until you achieve the desired distortion. This method can easily produce a lot of feedback, so keep your postgain or master volume levels lower initially before trying to get your full desired volume.
Use a megaphone, which naturally produces very distorted sound. You can buy a megaphone from RadioShack for about $100 (as of July 2011). Sing through the megaphone into your mic as you would sing normally, but be sure to turn the volume on your microphone input or PA down accordingly, since the megaphone will be a lot louder than your normal voice. The distortion produced by a megaphone will be extreme but can be very effective in certain situations.
Plug your mic into a distortion, overdrive or fuzz guitar pedal. The kind of sound that these pedals produce will vary drastically depending on the model you use, so experiment with different kinds and different settings. Some popular pedals for distorting vocals include the Ibanez Tube Screamer, the Boss SE-50 or the Pro Co RAT Pedal.