The difficulty of finding a teacher for your child depends on where you live. If you're in a larger city, you'll find many more teachers available to you.
You can find a teacher from several different places. Most parents will often start by asking the orchestra teacher at school where their child is going.
Obviously, if you know other parents of Orchestra students then you can also ask them. The best source there, though, will be the orchestra teacher.
Another good place to start is the local music store. Often they will keep a list of local music teachers. Keep in mind that even though they keep the list, that does not mean that they check the quality of each instructor. Occasionally they will be able to offer opinions, but you should always be willing to call around to each teacher to get a feel for their style before settling on one.
Different teachers teach in different ways. Some were taught in specific methods, such as the Suzuki method. And your child may not click with the first teacher that you try.
When I teach violin, I typically structure my lessons so that the first 5 to 10 minutes is review from last week. Then, I'll work with them on pure technical exercises that can range from note-reading to bowing exercises to fingering practice, depending on their level.
Then, we move on to new techniques (if applicable) and music. If my student is in orchestra, we will work on their orchestra music and then move on to the solo piece they're working on with me. (Students should always have a solo piece; you never know when someone might want an impromptu performance!)
The final 5 to 10 minutes, if the student has been working especially hard, I will typically let the student pick what song they want to play. It can be fun to end a lesson with a duet with the teacher!
Different teachers teach in various styles. There are strict teachers, fun teachers, teachers who demonstrate, teachers who only advise students on what they need to work on, and so on...
It's tempting to just go with the cheapest violin tutor. Sometimes, it'll work out just fine. However, remember that more experienced teachers have experience in making their lessons fun and informative, and often have more resources (such as copies of music for your child).
Here are a few more things to consider:
teachers will offer some sort of introductory lesson, or at least
reduced rate for the initial lesson. That is because they know that not
all teachers suit every student.
Typically, you can call up the teacher and ask them what their studio policies are, and they will tell you if there are any introductory / trial lessons.
The best way to find a good violin teacher for your child is to ask around. Get peoples' opinions. Go to the music store and ask there. Chat with several teachers either on the phone or in person and get a feel for them.
Your child's teacher is important and will be more than just a "homework coach". They will directly affect how your child feels about playing the violin.