Air Conditioning - Best way to clean dust clogged evaporator?

I have a farly new central A/C unit, and the evaporator looks to be clogged with dust. Unit has a boars hair filter that I have changed/cleaned regularly but doesn't seem to have done the job. I am good with my hands, and have a basic understanding of how the system works. What would be the best way to clean the evaporator.
I plan on finding a way to upgrade the filter setup to try to prevent this in the future. I'm new to this climate and have not had to deal with Central A/C in the past. I don't want to call a service tech and be taken to the cleaners if this is something I can do myself.
Thanks for any help I can get.

'how' did you determine the evaporator is dirty?

removed filter, evaporator is located above filter, looked at surface of evaporator fins are covered in dust/dirt. looking substance, removed panel above evaporator, top side of fins are clean, tried blowing back through with compressed air, didn't work. tried vacuum on bottom side doesn' work. label on outside says to use only mild detergent to clean evaporator. so I'm "assuming" there's a way to do this without disconnecting the evaporator coil and having to go through all that goes with that operation, ie. evacuating and recharging the system.

might want to see if an actual 'a/c' guy will reply.
'i' used a pump up -1 gal. plastic spray tank( like you use to spray for bugs)- used some liquid cleaning soloution ( simple green-for cleaning siding)-then washed it with clean water from same spray tank.
there is an actual-coil cleaning soloution.but i dont know that 'you' can buy it.
make sure -power to unit is off!
might need a wet vac to clean up mess.