A blurry background can serve many purposes in a painting. It can force the viewer to focus on specific objects or people in the foreground. It can also offer a visual flair that the painter desires for her work. Painting a blurry background requires the use of a mop brush which has a loose, wide group of bristles similar to a brush used to apply makeup.
Dip the mop brush in your paint. Apply enough paint to cover the tips of all the bristles.
Hold the tip of your mop brush against the canvas and move your hand in quick, back-and-forth movements to apply the paint to the surface.
Move the brush around to apply background colors in the desired locations. Reapply paint if your brush runs dry. Don't worry about detail as blurry backgrounds lack fine detail.