Ok this is something familiar to me during every string change, so here's my take of it:
1. Tape the fretboard on both sides of the 1st fret with tape(anything will do i guess, but i use one with low adhesive quality).
2. Use brasso with a cotton cloth to rub/fondle/stroke/carress/rape the fret. Using the fingernail to scrape/dig/finger the arse-crack of the fret that joins the fretboard, this is where most of the slime and grime breed.
3. Use another dry cloth to shine/buff/wax/dry.
4. Pull off tape and re-use for the next fret.
5. Repeat for all frets.
6. Follow up to oil fretboard.
Ps: Remember to remove strings before doing all of the above, i've actually seen someone trying to polish the frets with strings on and trying to go in between them to get the job done. What is wrong with ppl these days!? Enjoy.