There are books out on making t-shirt quilts. Check your library or Amazon.
As others have mentioned, you need to use a fusible to stabilize the t-shirt fabric before you cut and sew. I like fusible nylon tricot for this. Some fusibles make the t-shirt fabric too stiff, so be sure to experiment before making a final decision.
Usually the t-shirt logos are cut into squares or rectangles and assembled with sashing strips and borders (sashing and borders out of regular fabric).
Here is a click-link to a recent thread on the board. Scroll down the page to see the t-shirt quilt; this one was assembled without sashing but with "shadows" and is very striking.
http://www.quiltingboard.com/t-48453-1.htm