Lift the coleus plant out of the current pot. Pull as much soil away from the roots with your fingers as possible, but avoid pulling on or damaging the roots. If the soil is muddy and clinging to the roots, rinse it away with lukewarm water so you can easily see the root system.
Trim off any dead or mushy roots with a clean pair of shears. Mushy roots indicate rot, which can lead to a whole-plant bacterial infection.
Rinse any old soil out of the pot. Wash the pot with a solution of 9 parts water and 1 part bleach, which kills any bacterial pathogens on the pot.
Repot the coleus in fresh soil. Set the plant in the soil so the top of the roots sit just beneath the soil surface. Water the new soil just enough so it feels moist but isn't muddy or soggy.
Trim off dead leaves with the shears. Remove any foliage or stems that develop darker lesions or soft spots, as these may have an infection. Dip the shears into the bleach solution after each cut so you don't spread bacteria to healthy parts of the plant.