I just did this routine service on the bike I recently bought and thought I'd run through the procedure, in case it helps anyone here.

First, get fresh brake fluid. Most bikes need DOT4 grade. Be sure to get the right grade (it will say what you need on the master cylinder cover or in the owner's manual). A couple of pints is usually enough to flush the entire system and refill with fresh fluid - which should be done every 2 years to get rid of contaminated fluid and maintain the system's function.

Get about two feet of rubber hose (clear is best but black is fine) in a diameter that will fit snugly over the bleeder nipple on the brake caliper.

Get a clear glass.

OK.

Now, carefully loosen the screws that hold the master cylinder cover in place. These are sometimes easy to chew up, so unscrew them with great care to avoid a big hassle.

Assuming dual discs, pick a side. It doesn't matter which one.

Next, pour some fresh brake fluid into the glass; an inch or two is plenty. You want just enough to cover one end of the rubber hose. The other end goes on the bleeder screw nipple, after you crack it open about 1/4-1/2 turn with a wrench.

With the bleeder screw open and the hose in place (one end submerged in the fluid in the glass; the other firmly on the nipple) gently pull the brake lever in and then out. Repeat. You will notice the reservoir gradually emptying as you do this. Don't let it empty all the way (uncover the piston in the bottom of the reservoir) or you will get air into the system and have to start over. When it is nearly empty, refill with fresh fluid and repeat. Then close the bleeder screw and switch operations to the other side/caliper. Repeat the process.

This method will quickly purge all the old fluid (and any air in the system) in just a few minutes, without the need of an assistant - and without making a big mess. The glass of fluid/hose keeps the old escaping fluid under control and prevents any air from getting sucked back into the system - which would probably happen if you didn't have a helper to manually open and close the bleeder valve as you worked the lever.

This is the method I have used for 20-plus years and I have always had good results.

Hope it is helpful to someone!

PS: The same method works equally well with cars, too.