A google search will tell you all you need to know... there are tons of discussions about this.
F-13, Purple Power, Simple Green, Krud Kutter, LA Awesome, are all chemicals that people report success with, along with several others. F-13 and Purple Power are pretty popular. If you cant get them clean with your gutter brush, magic erasers might do the trick.
People apply by directly spraying the gutters, by spraying the brush and applying, or by dunking the brush in a 5 gallon bucket of mix and applying. Most apply the chemical, let sit a few minutes, then scrub with brush, and rinse.
I've seen people report charging anywhere from $1 to $2 a linear foot, depending on how many stories and other things.
Dilution rates vary per product. Start weak and work your way up, these products will eat the paint off of gutters if too strong... warmer temperatures make them more potent. Purple Power maybe start 1/2 and 1/2 and work your way up.
I've PM'd you a link to a thread that discusses F-13 and Purple Power, as well as detailed application techniques.