Finding the right teacher is important.
While personal achievement has it merits just because someone is good at something doesn't make them good at teaching that same thing. Are there core characteristics one should look for? What kinds of questions should I ask to find someone that is right for me?
Positive energy and attitude. You want to find an instructor who is there, and doesn't have his upper belts teaching all the time. Sit in and participate in some classes, make sure you feel comfortable with the school and how the other instructors handle themselves and treat students.
Too many times I've walked into a school and it's be constant yelling at the students or other behavior not suited for real learning. Martial arts is a journey, your instructor should still be learning as they teach you.
I always find it helpful finding out why the instructor got into martial arts, what drives them to be a martial artist, why do they teach and how are the continuing their martial arts journey.
Most important, make sure your comfortable with the school and all the instructors. If your not comfortable, or can't get comfortable, it won't be a positive or fulfilling experience.
He should have great students.
That is: you realistically have more chance of becoming like the students of a teacher rather than like him/herself. There are plenty of high-ranked senseis with bad students out there. They are very good at practice and have a great reputation, but apparently they don't get good results at teaching.
Are there core characteristics one should look for?
Affability.
Quite simply, we learn more from teachers we like. We like people who are like ourselves, or rather, like the people we wish we were. Simply put, if there's something that we find particularly lovely about another person, we're more inclined to do as they ask and find interesting the things that they say.
This works well for martial arts too since we're asking someone to both invade our personal space and that they respect the trust we're placing in them. There is, of course, a level of intimacy that goes along with this, which is why most people who genuinely like the school they're in spend a great deal of time there, or with their instructor or classmates in social situations.
Secondly, your interest in what he's teaching. If you have no interest in the subject matter, no amount of interest in the sensei will change that. This can be as grave as you simply don't find the art interesting, or as simple as his approach being sport while your interest lies in personal defense.
Thirdly, a knowledge of the material. It's rather a common con to see an instructor with lots of framed certificates teaching something that's passably some art, or would be to the untrained observer. If you take the trial lesson or sit in, observe, and attempt to remember the format that a technique took, then compare that with known high-quality instructors of whom video evidence exists (Youtube can be your friend!), you can determine at least reasonably if they appear to know what they're doing.
What kinds of questions should I ask to find someone that is right for me?
The problem with questions is two fold: if you do not know the answer, you simply need to know if they're lying. If you can not do either, then the question serves no purpose.
The basic questions that you should know are:
Anything else can be helpful, but these should be your priority.
Non-toxicity
No one thinks of this, until they've been burned by it. A lot of people come to martial arts out of an emotional insecurity - fear, the need to feel control in something. You'd think once they become adept in their art, they'd get over it, but a lot of instructors will simply inflict those same old problems onto their students.
I've seen instructors who emotionally abuse students, sexually exploit students, set up interpersonal drama between students to keep them from "outshining" the instructor, etc.
I look for three things to avoid this:
Do they have a mean streak? Do they have nasty insults about people? About the students in front of them?
Do they listen? Do they actually respect requests from students? Or do they simply say "try harder!" regardless of the situation?
Are they relatively straightforward about how they learned what they learned? Is there at least enough humility to admit they've received training from others? They don't have to worship a particular system or teacher, it's just that the instructors who seem to erase the fact that they, too, had to learn from others, tend to like to set themselves up on pedestals.
Basic presentation to specific advice
The best way to see how well you'll learn is to train with someone in a tester class or workshop. You can see how they present the basic material in demonstration, and how well they give specific advice to each student.
Not everyone is good at both! Which means some instructors might be able to give you the general gist well, but you'll have to figure out the smaller details as specific to you, through investigation and hard work. Other instructors might be terrible at explaining the general thing, but if you get one on one time, you can really learn a lot quickly.
Having a question or two between learning things is a good way to see how well they do the specifics - "I'm having a hard time doing X..." "I can't seem to get the power into this technique..." "This feels weird, is it because my arms are too long/short compared to my partner?" etc.
Adaptation
Any teacher can show you exactly what they learned. A good teacher can show you what they learned in a way that is better for you, specifically.
This means adapting techniques based on your body type (tall, short, heavy, light), injuries (knee, hip, back issues) and most importantly, being aware of what you need to prevent injuries.
One of my best kinesiology instructors also teaches acrobatics work and he points out that a lot of martial arts will demand people increase flexibility without looking at the person's body. Not everyone's bones grow exactly the same - some people literally cannot get greater range of motion in some joints without sawing part some of their bone structure. Other people may have joints prone to injury and what they need is stabilization and strengthening, not loosening of those joints.
Instructors who know their art well enough to help you train safely based on your needs, including the needs you may not have known but they can see? Those instructors are well worth their weight in gold.
I agree with the first two comments, is there instructing and has great students.
If you see someone who is third degree black belt but looks sloppy in everything they're doing you may want to look else where.
More importantly you want to find a school that doesn't just let you test every two months. Those schools are just trying to make the extra money earned from tests. If you witness a student ask the instructor when hes going to test and the instructors response you will test when you're ready/don't ask me about testing when this is still sloppy/30 pushups for asking me...then find a different school.
Generosity. Good teachers want you to "catch the bug" and make the training your own. They want you to fall in love with it. To get you to that point they will go out of their way to support your development. If the teacher doesn't have a giving spirit, it will show. They'll concern themselves only with the more experienced students. A good teacher will find a balance between helping the strongest and weakest of the students and make time for anyone who asks.
Excellent answers by @swift and @tacome; I believe they are correct, particularly with respect to affability; this person will be motivating you, it is vital that you respect and like this person.
I'd add:
Flexibility. My teacher has split with one other teacher because that teacher insists that every technique must be done exactly the same way. There is simply no way that I (six + feet, 300 pounds, weak knees) am going to take exactly the same number of steps to unbalance someone as my daughter (5'2", 100 pounds, astonishingly flexible). The core technique remains the same, but each person is bringing a different physique to the party. I'd be cautious of any teacher who refused to permit any variation. Special variation - which has been cited many times on MA:SE - medical flexibility. If the teacher can't flex to accomodate injury/weakness/medical/whatever, then find someone who will.
Respect. My teacher split with another teacher because she asserts
that she has mastered everything that Japan has to offer. There is no
reason for her to train in Japan because she is that good. But I'd be
very suspicious of any teacher who can't tell you what they're
learning and who they're learning from. They get partial credit if
they admit they're learning from their students, but everybody has to
look up to someone. If the teacher trash talks other styles, I'd
have trouble learning from them. (be rational; my teacher has
expressed reservations about the efficacy of certain techniques in a
fight situation. That's different from trash talking.
Scam detection :
Unfortunately there are many so-called masters out there, that have rather creatively found ways to rip off students. These methods can be subtle to very obvious. I'll list out some that I've actually come across :
Martial Science
Random
Reply to the comment by Tomas (clarifications)
Maybe they should, I can understand if the science bit is left as an excercise for the student to figure out - so that spoon feeding is avoided. Or if this is taught at a latter stage in order to avoid confusing those who are new. I personally feel that you can learn much faster and avoid injuries, by learning why a technique is performed the way it's performed. Details do matter, especially when you look into branches like Wing-chun, where every move is designed to have an edge over the opponent in every parameter of fighting (speed/timing/position/damage inflicted/centre of gravity).
teachingandrecommendationtoteaching-recommendationwhich seemed to have more appropriate meaning for this question.teachingseems to imply the question itself is about teaching.