Since coleus is grown for its foliage, not its flowers, use a water soluble fertilizer higher in nitrogen than phosporus, like Bachman's Excel Gro Acid formula. If you are using an organic fertilizer, top dress with blood meal in mid-summer. The new varieties are mostly self-branching. Seed strains will need pinching to prevent blooming so the coleus will be longer-lived. When you see a flower head forming, just pinch it off above the next set of leaves.
Coleus do very well from cuttings.
Water the cuttings well. Use a humidity dome or enclose the pot in a clear plastic bag to minimize evaporation. Keep the cuttings in high light, but out of direct sunlight. In about two weeks the cuttings will have rooted. Remove the dome or bag and grow them as a pot plant for the winter or plant them outside in summer.