Vinyl wallpaper is far less porous than wallpapers made from paper or fabric. Because it's vinyl, it's easier to install and cleans up with a damp cloth. But like other wallpapers, vinyl wallpaper can be difficult to remove. If you want to paint your wall without removing the vinyl wallpaper, you need to alter the integrity of its surface. You need to prepare the vinyl so that it will accept both primer and paint. Otherwise, the paint will "thin" and the wallpaper will show through.
Wash the surface of the wallpaper with a degreaser. Clean it with a cotton cloth or sponge and rinse the degreaser off using clean water. Dry with a cloth and let dry for one to two hours.
Score the wallpaper. Scoring the paper will break the surface tension. You can do this with a commercially available product such as Paper Tiger, or you can improvise and use a ravioli cutter or similar tool. You need a razor-edged roller to roll over the paper. Use broad strokes; you do not need to score every inch of the surface.
Lightly sand the wallpaper with fine-grit sandpaper. Do not use a sander. Just as you did to score the surface, use broad strokes, just enough to break down the surface. Wipe clean with a cotton cloth.
Apply a coat of quality primer to the wall. If you are painting the wall a dark, rich colour, tint the primer to a shade two or three shades lighter than your paint colour. Allow the primer to dry for 24 hours. If you are applying a light-coloured paint and the wallpaper is dark, apply a second coat of primer for better coverage.
Apply a coat of quality flat or semigloss paint to the wall. Use a short nap roller to apply the paint, as the wallpaper surface, even with the primer, is more like a smooth wall than a textured wall. Allow the paint to dry for 24 hours before applying a second coat.
If after the second coat, you still see wallpaper, apply primer tinted to the same shade as your paint colour to the wall and then apply a coat of paint.