It depends on how many teachers are there. If it's a language with not many teachers, I might go through all available teacher profiles and check who's in my price range and can offer classes at times that are convenient for me. If it's a language with many teachers, like Spanish, I also first use the filter for price range and availability, maybe country of origin (I had classes with Latinamerican teachers later, but in the beginning I was looking for teachers from Spain). Then I scroll through the profiles and yes, it's a bit superficial but some teacher avatars are more appealing to me than others. I'm not looking for people who are attractive, but sometimes I like a smile or there is just something about the teacher that sparks my interest. Then I watch the video, read the introduction, look at the student/class ratio (are students coming back?) and read some of the reviews. I have to admit that I often close the video after only some seconds if the teacher starts with "hello everyone" instead of talking in my target language. But that's just me. I'm not interested in listening to the teacher talking in English but I want to hear them talking the target language. I'm bored when the teacher says the usual "I know how hard it is to learn a foreign language". What I'm interested in are the methods and materials that the teachers uses to teach the language. When I like the video, and price and availability fit, I book a class. I was rarely disappointed with my choice. I also did book classes with teachers who corrected my notebook entries or gave me valuable advice in my discussion threads.

Pretty much the same process as Miriam's, though I like to go to tutors who don't have so many students yet. My schedule might change quickly so I like to schedule the classes a day or two before the class. This is easier with with not so fully booked calendars. I also like that I can support new teachers like this as they need students and reviews to start getting momentum. I'm also mainly looking for conversation practice and informal tutoring.

Your writing suggests that you are looking to have only one teacher? Is this right? You can have several and that would also be better because you would get exposure to different ways o talking and you can then talk about one subject with one teacher and then later try the new words or grammar with another teacher and see if they understand you or if they have another perspective to it.

After reading Petteri's comment, I'd like to add that the return rate of students is only an important factor when it's a teacher who's been around for a while. If a teacher tought 100 classes and had 100 students, then there's something off. But I don't mind being the first or one of the first students of a new teacher if the profile is otherwise interesting. Everyone's got to start at some point.