The first step is to find an artist that already paints the genre that you are interested in. For example, if you want a portrait of your dog, then asking an artist that has never painted animals is not going to work so well. You want to love the painting, so the artist needs to have skills in that area…demonstrable via their portfolio of work. Secondly, their painting style is important. I would say, and may be shot down by other artists, that you can ask a fine artist to paint in a more abstract fashion, but not always the other way around. Realistic fine art is a skill separate from other styles and requires years of dedicated self-practice.
Once you have found an artist, look to see what prices they have been charging for their work. If they are selling pieces for £500 they will probably be offended if you ask for a painting for £50. Artists work hard to get their value high and they do this by growing their popularity and demand. You are paying for an artist to sit for hours to create your piece. Even at minimum wage an 12-hour painting would cost you £98.52 next year (2019).
Once you have style, price and genre sorted. You will most likely need to gather some photographic sources of reference for the artist to work from. They will tell you what they need. Then you are all set. Just make sure you allow plenty of time for a first view, to the final piece.