You should probably go to a good place which will give you a good grounding in philosophy. I don't think it is a good thing to decide from the outset narrow field you want to study. You may find you will change your mind. In any case, from a practical standpoint, you are going to have to take general examinations in the various fields of philosophy: logic, epistemology, and so on. So you are going to have to prepare for those.

I'm sure you could focus on existentialism at most universities with a healthy philosophy department. You should look into any professors in the field you'd like to study with and seek them out.

If you want to become a tenure-track professor at a university, you should basically get your Phd from the most 'prestigious' (highly ranked) program you can get into, all else being equal...

In one sense, at the graduate level, one can specialise anywhere, provided the faculty has one or two professors with a grounding the in the subject. In another sense, your educational experience will enhanced if there are major figures in that area lecturing at a particular institution.

Obviously it is not the case that all Existentialists teach at one or two universities. As you have undoubtedly kept current in the literature and journals of philosophy, you should be able to identify several places from the number of writers in the subject. Again, your own professors may be able to suggest departments with a strong Existentialist bent, and a review of the catalogs from these will also help you to make a determination.

For the most part, I would think that more often than not, you will not find "Existentialist" faculties, but rather those with professors who specialize in particular "existentialist" philosophers.Your own predilection for the writings of one or two of them may thus be a deciding factor.

From general to specific areas, then, try to find faculties with strong representation in:
Modern Continental Philosophers
Phenomenology and Existentialism (you may have to read between the lines in departmental catalogs)
One or two writers (e.g. Nietzsche, Sartre, Heidegger), with courses devoted to their works.