Ultimately you want to show a few things with a portfolio of this kind and in about 10 - 20 pieces.
1. You should not only show that you are passionate about learning new things and that you love graphic design, you should be passionate about it too. This, of course, can be written in your covering letter but by showing a variety of different designs (like logos, posters, album covers, some illustrations etc.) will show you are not afraid to try different things and are open to learning.
2. Next, you should show some technical ability and understanding of a few of the programs you would be using. So a bit of Photoshop work where you've played around with layer properties and effects, some Vector work in Illustrator and maybe an interior layout, will help.
None of this has to be 100% polished because you're not looking for work, you're looking to get more education in the field. By showing willingness and some ability, you should communicate that you won't be a waste of time to teach. Now, this is what I would a) look for in someone I was going to teach and b) what I would do if I was looking to go into higher education.
I hope these help. If you want any more help, email me some links to an online portfolio and I'd gladly take a look at your work to give you some more pointers.