Do you mean an existing black background such as a wall or projecting the background. Is it a slide projector, video projector etc. Different projectors may require different methods
If it's a true black background, what you propose is impossible. The definition of a black surface is one which absorbs all visible light. So,no matter what you project onto it, it won't be reflected back.
There may be alternative solutions if you can change the colour of the background. (Please post again and we'll try to help you out). But if the background is black, no light can reflect from it.
As Buenchico says, projecting onto a black background won't yield very good results. Think about the last time you went to the cinema and there was a black screen.
To project a red cross using white light you will need a red filter and a cross shape cut out from either card or thin metal, although as Mortartube says it depends upon the type of projector you are using.
The black background simply absorbs the light. This is also creates a problem when trying to obtain distances to anything black using laser measuring devices. There often just isn't enough reflected light for the system to work.