First of all, you need to print your character on a iron-on transfer sheet. You can buy this from your local bookstores or even online.
Make sure you print it with a "reverse" setting since you will be transferring the print flipped (you don't want that).
Use an old shirt you wouldn't mind tearing apart. Cut out the character and iron it on. You can use a clean sheet of paper on top of the cut-out so that it will be protected (just in case the iron will be too hot!).
How about turning your old white tee to something more interesting like a DIY graphic shirt? You don't need to have your design printed somewhere expensive + sometimes, fabric pens are too tacky to use.. >_< Whenever I need a customized shirt for a presentation (back in college), I would use iron-on transfer sheets to make personalized shirts.
Now you can proudly tell people (when they ask you about your shirt) that it is your design!
First, you need iron-on transfer papers. I suggest getting an A4 size if you want a huge print on your DIY shirt. You can print many designs in one page then just cut those out separately to save the paper.
You can buy iron-on transfers in eBay, amazon, stationary stores / bookstores..