The Painter achievements are much easier to get than the Builder or Gatherer achievements. Painting is done by making paints from flower seeds in the paint mill and using the paint brush with the dye on blocks of your raft.
The base flower colors you can find are black, white, blue, yellow, and red. I recommend only painting with the base colors as it uses the least amount of paint. I also recommend painting blocks, using the erase feature of the brush to clean them, then repainting the blocks, rather than painting over already painted blocks. Erasing a painted block counted towards the Painter achievements in previous updates however I am unsure if it still does.