The process of commissioning an oil painting, sketch, or drawing to immortalize your beloved pet is actually very simple!
Pets usually aren't so good at sitting still long enough to be painted, so a good second step is to round up as many photos of your pet as you can. The more options you give me to work from, the happier you will be with the end result. Depending on the photos, I can help recommend size and medium.
One more thing to consider is where you want to display the portrait. Where you want to place it will usually help you determine what size artwork you'd like.