If possible, I like to screen print. Air dry screen printing flocking adhesive (Nazdar PX) will work, as does plastisol (curing the plastisol WILL shrink the fabric considerably). I use heavy body acrylic paint thinned with a bit of acrylic medium (kind of a one shot deal, as the paint will dry fast and be impossible to remove from the screen - mix in a small amount of KY jelly if you want to extend the dry time).
If you don't want to go the screen printing route, you can use acrylic artist paint and hand paint the graphic. Multiple thin coats works best; it will take a few coats to get full coverage. If your freehand skills are dodgy, you can use a vinyl stencil. I press down the edges firmly with the corner of a squeegee, then apply a coat of clear acrylic medium to the surface first. When that is dry, I paint with full strength acrylic with multiple coats until I get the desired coverage. When the final coat is dry to touch, lift up the stencil carefully. The initial clear coat will "seal" the edges and help keep the pigmented paint from bleeding into the surrounding surface.
Mostly these days I just order the awning with graphics pre-printed. The large awning wholesalers are set up to do this (they will typically screen print - with materials at hand they can make up a screen in about ten minutes). I have also successfully used digitally printed fabric for short term applications.
https://www.sailrite.com/PDF/Sunbrella-How-To-Apply-Graphics-To-Awning-And-Marine-Fabrics.pdf
