I'm not involved in the graduate admissions process in the Harvard Government department at all, although like all of my colleagues, I receive a large number of email queries about the application process from prospective students.
In case it's helpful, then, a couple of pieces of advice specific to the graduate program here at Harvard, before a couple of more general pieces of advice from friends and colleagues in the discipline:
You may wish to consult the information posted on the Government department website, which has more details about the program and admissions process, as well as descriptions of research interests of current graduate students and faculty members.
Unlike PhD programs in other disciplines/countries, our admissions process does not rely on faculty sponsors: that is, if you’re admitted, you’re admitted to the program more generally, rather than to work with a specific faculty member. In this sense, because the admissions process is centralized, and there aren't particular slots reserved for particular supervisors, you don’t need to go through the hassle of contacting individual faculty members to tell them that you're applying, or asking them to sign off on your application or research interests. Similarly, the sheer number of applications we receive (in a 1993 PS article, King, Bruce and Gilligan report the Government department was receiving ~700 applications a year) means that it's logistically impossible for us to meet with all of the prospective applicants who might wish to do so. The good news is that because the admissions process is centralized, this isn't something you need to do anyway! Once you've been admitted to the program, we'll fly you in to meet with us, so you'll have ample chance to meet with us then.
Finally, a number of friends and colleagues in the discipline have put together some helpful resources with advice about applying to PhD programs in political science in general, and in international relations in particular:
"Should I Get a PhD?" is an interview-based site run by Tim Hopper that isn't specific to political science, but offers lots of helpful suggestions more generally about the more fundamental question of whether you should apply for a PhD in the first place.
Dan Nexon has helpful advice at the Duck of Minerva on applying for a PhD in political science, and how to make your application more competitive.
Nuno Monteiro has great advice both on how to decide whether to go to graduate school, and the tradeoffs between PhD programs (like those offered by the Government department) and MA programs (which the Government department doesn't currently offer as a standalone degree).
Erica Chenoweth has similarly helpful advice; her discussion of the difference between policy-oriented degrees (like those offered at the Harvard Kennedy School) versus academic degrees (like those offered in the Government department) is especially valuable.
Steven Wilkinson has useful advice especially relevant for international applicants.
Terri E. Givens has a series of helpful articles at Inside Higher Education on her graduate school experiences, especially relevant for first-generation students, and students of color.
Duke's Sociology Department has a helpful FAQ page that's technically about applying to sociology PhD programs, but many of its suggestions apply to social science PhD programs more generally.
Dan Drezner has a series of helpful posts at Foreign Policy on PhD programs in political science: see here for advice for undergraduates, here for advice for students who have already graduated, and here for advice on PhD applications for aspiring policymakers.
Erin Simpson and Andrew Exum have helpful advice on the CNAS blog from the perspective of policymakers.
Bradley Potter, Nathaniel Allen, and Torrey Taussig have helpful advice at War on the Rocks about good and not-so-good reasons to pursue a policy-oriented PhD.
Chris Blattman has extensive advice on many of the above topics from a political economy perspective.
Cyrus Samii has helpful advice on much of the above; his advice about applying to "boutique" departments is also particularly helpful.
Justin Esarey has a helpful post on The Political Methodologist about how to know whether to apply to grad school in the first place, and if so, how to choose where to apply.
Once you've been admitted to a PhD program, Chris Kennedy has useful advice about how to prepare for your first semester.