How to establish a relationship with a department head as an undergraduate?
It's possible that you are misunderstanding what the signature of a department head implies or requires. In general, signatures are used to ensure that policies and procedures have been appropriately followed. Save for unfortunate cases where resources are solely allocated due to rather extreme corruption (beyond anything I've personally encountered), you don't have to have some sort of deep personal history or relationship just to take part in an existing publicly advertised academic program.

The most common way this comes up is a form that requires two signatures - one from your advisor (or other faculty member relevant to the request), and one from the head of the department. The advisor/faculty member is usually the one responsible for making sure you meet all the program requirements, and believes you should have the request indicated by the form authorized. Then the signature from the head of the department is generally just a bureaucratic formality, and the department head's job (often delegated to an assistant) is to ensure you qualify, etc (making sure the faculty member did their job correctly), maybe ask a few questions or require supporting paper work if applicable (if it's routine they won't even do that), and then sign. And off you go!

To see what is truly required to get into your school's program, you'll need to look up further information about it on the website - if one exists (it may not have useful info), and talk with whoever administers the program. You might also just go to your department office and ask the assistant for more information about the program. You can just say, "I read about the program where you can work towards a masters while you are an undergrad - how can I find out more about this program?" 

You can then find out if the program is a competitive application process, are there certain qualifications you need before applying to the program (such as completing a certain number of courses, having a certain GPA, meeting with a graduate advisor, etc), and all the other details you might want to know about to decide if the program is right for you and how you can join. It could be as simple as ask to join, give your transcript to the assistant and they will have the chair sign, and you are in (so long as you pay the bills, which might increase to be a part of that program). But you won't know until you talk directly to the appropriate staff members - no need for scheming just yet!