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You may use raster images as backgrounds, or you may have a look at the effects available in your version (the Effect dropdown list is/was arranged with vector effects at the top and raster effects at the bottom); there may be more backgrounds in the same artwork.
Depending on the desired (combination of) effects, you may use Clipping Masks, Opacity Masks, Knockout, or compound Paths to get the grungy parts to show. You may look up these, and how they work, in the Helpfile.
For more explict suggestions, you may also post in the Illustrator forum.
The right resolution of the raster image or raster effects (Effect>Document Raster Effects Settings) depends upon the printing process and the relative size (combination of size and intended viewing distance). The long answer is to ask your printer.
Try doing a search using keywords weathered texture effects or similar and you'll find some interesting tutorials. I wanted to create weathered texture on text elements and found two tutorials: 1.) vector; 1.) bitmap. You can do it all vector, but it will require a coversion from bitmap to vector. Interestingly, the bitmap tutorial suggested the effects be done in low resolution and then sized up later ( this gives you added roughness in the appearance ). The Photoshop method is a 4 or 5 step process using filters. Vector has the added step of converting from vector from a bitmap texture file.