The Sims 4

There are 10 levels in the painting skill. Painting is one of the easiest ways to make money in the Sims 4. Each painting can be sold or kept. This skill goes along with the painting career, so you can max out both the skill and job in one go. The lot trait Natural Light will give the sim a +50% skill gain boost and the lot trait Home Studio, will help the sim make higher quality paintings at a lower level.

To get started either buy an easel from the buy mode or purchase the painting skill books from a bookshelf.

I found the easiest way to get this skill maxed out is just to have a jobless sim do nothing but paint in their free time. It helps fund the household while also getting an achievement done.

The guide Lymiss wrote is very helpful but I wanted to add a few things in:

You can do this alongside the Painter Extraordinaire aspiration to gain some reward points - this will let you buy perks and such such as instant refilling of your needs, or further perks such as Savant which allows you to gain skills much quicker

You can also do this alongside the Painter career which will increase all your money gain as you go.

To make this easier, and what I did, was used the Digitalistic Sketchpad. You can buy this under the Electronics tab in the store for 1,500 Simoleons. Pause your game, and fill your queue with paintings before speeding time up again. I found this worked much quicker than using the easel, as one set of queued paintings did about 1/4 of the skill bar at higher levels. I didn't use it at lower levels and began using it at level 8, but I imagine it will speed through the small level bars much quicker than using the easel.

Art your Sim makes on the sketchpad will be auto put into their inventory. You can sell it by dragging each painting to the easel and then selecting from there, or you can use the Street Gallery item and sell your paintings on it for a mark up price in a similar way to car boot sales to make more than selling from the easel.
(Remember, there's also an achievement for gaining 1 million Simoleans in one singular household: )

Personally, I had one Sim working and another jobless Sim work on the skills to max them out, which ensured I had a good income coming in, but others have said playing the Aspiration and Career alongside this helped them max out as well if you are playing with a one Sim household, or also trying to get the painting career achievement: