How does a faculty member seek administrative responsibilities in a Department? Do such positions come in a cycle only? Or should the faculty member make his/her willingness known in advance to the Chair? Are there any politics/wrangling involved?
Moreover, for an assistant professor, is there anything that could be done during PhD so that the profile comes out as someone willing to take admin responsibilities?
Don't worry, administrative responsibilities will find you unless you flee them, and even then you probably won't avoid them. At most universities, young faculty are expressly spared administrative responsibilities so that they can focus on research and get tenure.
If you're really looking for them, generally all you need to do is vocally speak your mind on every issue that comes up. Most of your colleagues will feign apathy in order to avoid being assigned to a committee.