Is it possible for someone to get into a Ph.D. without a thesis at Masters-level?
If someone doesn't have a thesis at his Masters-level, what other criteria can he fill in order to approach a potential supervisor?
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I assume that you have a Master's degree but not a Master's Thesis. The answer to your question depends on the location.
In the UK and the US, a Master's degree is not a requirement to start PhD studies, and therefore a Master's Thesis is also not a requirement. The admissions committee will compare your achievements with those of other students who only have a Bachelor's degree, and the expectations in terms of research output and thesis writing will correspond to that level.
In other locations, such as Germany, starting doctoral studies without a Master's degree is possible but very rare (unfortunately, the link is in German, but it says that only 1300 out of 108000 doctoral students that started on the winter semester 2011/2012 had no Master's degree).
The traditional way of starting doctoral studies in Germany is to contact a professor directly, and be selected by the professor, without the participation of any admissions committee. Professors will generally expect a Master's degree, and will want to have access to some of your research output to see how good your research is. They will by default look at your Master's Thesis for this, but you might be able to also get selected without a Master's Thesis if you have enough research output (publications or written reports of research projects).
There are also structured doctoral programs in Germany. These function in a similar way to PhD programs in the US/UK (with admissions committee and more course work), and are more likely to accept students without a Master's degree or thesis. These programs are, however, much less common than "traditional" doctoral studies as per the previous paragraph, so you might have a hard time finding one that interests you.