Your question asks how to create a custom list model, however, that's not what you need (I don't think) as a DefaultListModel will work nicely for you. Rather you will need to work on the renderer. You need to create a non-GUI class to hold your information that each item will display, probably your Contact class, and then create a JList that holds this in its DefaultListModel.
The key for you will be to then create a custom list cell renderer to display the information on multiple lines -- perhaps a JTextArea, or a JPanel that holds two JLabels in a GridLayout. Please understand that the renderer does not display the actual underlying components, but something more akin to a stamped image of whatever components you're trying to display, so it will not have the full behaviors available to it as the actual component would. It will take work, but the writing a renderer section of the tutorial linked to by user714965 will show you how to do this.
Please give it a try, and then if you still are stuck, come on back with your code, your errors, and your questions, and we'll be better able to give you specific help.