Just don't.

The coating on the surface of your camera's mirror is the most fragile piece of your entire camera that is accessible without taking the camera apart, probably followed closely by the underside of the focusing screen located just above it. The mirror should never be touched on the surface. Unlike most mirrors in other applications that have the reflective coating on the back of the glass, the mirrors in cameras have the coating on the front surface and it is easily damaged. At most, use an air blower to gently knock dust off of the mirror.

Trying to clean the mirror will likely result in scratches that are worse than the dust you wish to remove. You can also easily wipe part of the coating off, reducing the reflectivity of the mirror. Anything that changes the reflectiveness of the mirror or the transmissive properties of the focusing screen can also affect the accuracy of your camera's light meter, which is located above the focusing screen. The light measured by the meter must first reflect off the mirror and pass through the focusing screen.

There are a few specialty service providers that can use a special chemical process to clean the mirror, but it is expensive even before the shipping charges and insurance add insult to injury. Even many factory authorized service centers will replace (using very delicate handling procedures) rather than attempt to clean a reflex mirror in the camera's light box.

Any DSLR camera that has been used for any length of time has a little dust on the mirror. The best way to deal with it is just to ignore it and keep on shooting. After all, even factory fresh lenses have a little dust in them as well. Unlike the mirror, that dust is actually in the light path of the image as recorded by the camera.