Here are some basics that can play a role and improve one's chances, from a U.S.-centric point of view:
Your postgraduate degree is reasonably recent (preferably immediate prior year or close, with relevant experience in between). If there was a lengthy break with seemingly unrelated work in between based on what you want to study now, this complicates things.
You are applying in the same general discipline as your postgrad degree (e.g. Marketing, Mechanical Engineering, Psychology).
You had good letters of recommendation. If they had any flags that might give cause for concern for admitting department, that is a problem that you need to iron out prior to sending out any more applications. If you have reason to think this may be the case, it may be best to approach your recommending faculty honestly about this and see if they can either share a copy of the letters they wrote for you, or tell you if they mentioned anything that might explain slow uptake.
Communication with faculty:
- evidence of "passable" English proficiency (if non-native), or (if native) ability to write in pretty much perfect English with no grammer or, punctuation errors.
- Have all your applications been "cold," i.e. with no personal connection to / past interfacing with the faculty whom you contacted? In other words, have the letters of recommendation (which I am assuming
If so, this might explain reticence to respond. The Academe is a fairly tight community of several thousands or hundreds of individuals in any given narrow field. The "first order" professional networks tend to be on the order of a couple dozen past/present collaborators. Faculty tend to be more comfortable receiving a recommendation from someone they know rather than a complete stranger. So ask your professors who write your recommendation letters for contacts they would recommend, and tackle those programs as priority #1, gradually widening to "outer rings" of your recommenders' professional networks.
Maximally polished, edited, proof-read, 100% perfect letter of interest (cover letter/whatever they call it). The letter must show evidence that you have done your homework and are aware of the latest research projects of the faculty whose work most closely fits your experience and interests and who you would like to advise you.
High score on GRE or other applicable standardized graduate entrance exam.
- As has been suggested elsewhere on this forum, I would avoid stating anything about the fact that you are not seeking funding. This self-qualification might come across as, "I am hoping the fact that I am a cheap date will sweeten the deal." This might work in some odd cases, but most reputable, large research universities tend to be more concerned with attracting high-quality applicants rather than affordable applicants.
If the department considers the applicant to have potential and leans toward admission, but there is no funding, they will communicate this though an admission decision without corresponding offer of Research or Teaching Assistantship. This leaves the ball in the applicant's court, and things work out through natural attrition. (For instance, I had no offer of support first year, so found a Research Assistantship with another research center on campus that was closely related to my field, and it ended up a good fit.)
- Have you met, or at least read / cited in your thesis or research papers the professors you are contacting? Professors like applicants who do not only talk the talk (in a cover letter) but walk the walk, having used their theories and cited them accordingly in written work. Naturally these pieces also make for good writing samples, if the department and/or the Graduate School requests those as part of the application.
Finally, basic psychology suggests that applicants who behave like pests which, if not kept at an arm's length, would swallow all of a professor's free time and clutter his/her inbox, are less attractive, than applicants who communicate no more than necessary as part of the application process, and follow-up only with good cause and after first trying to get answers with the Graduate School and/or department's admissions office. Professors value their time immensely, and aggressively prioritize what emails they respond and do not respond to. They are not trying to be rude, it's just a professional adaptation to an environment with multiple competing priorities and limited time. So you want to come across as someone who recognizes this and helps them spend less, not more time addressing your inquiries.
UNLIKE THIS EMAIL, if you do email faculty, keep your email inquiries SHORT: 3-4 sentences, and get to the point in the first or second sentence. Begin with a question and request, and end with context. Professors often do not read to the bottom of every message.
Communicating through the appropriate administrative channels and only when necessary beyond the formal application process makes you appear confident, and confidence may be perceived as evidence of having other options. In a sense, you want to balance being courteous, excited, and a little reserved at the same time.
Finally, I would encourage to consider looking closer to home. Why apply to all continents if there might be a great program a few hours away in a neighboring town in the same country...but I recognize there may be other factors at play there.
I am sure I am missing some other important details, so extensions/comments welcome as always.