What Employers Are Looking For

First Impressions
The obvious place to start is with your CV. It should get the employers attention fast, ie. about 10 seconds, so make sure it's impressive. Try reading teaching CVs to find out more.
Perhaps more important, but often overlooked, is your cover letter. Personalise it, get the name of the person you're writing too and tailor it to each job. Don't press send too fast - you'll end up sending it Mr Smith when you're applying to Mr White's job advert. Find out more about cover letters by reading teaching cover letters.
Secondly, when applying via email, try putting this in the subject line: Your name, highest degree held, and years of experience. It will make it stand out more. 
Show Me The Money!
So if you've got a professional CV and cover letter, the next step is to know what you're worth. Forget about the whole cost of living thing. Don't buy what some employers are telling you about how you'll make two, three, or five times as much as the locals.
That's completely irrelevant since you're not going to live like a local. You probably won't speak the language, and even if you do, you won't have a network of friends and family like the locals do. Nor do most locals want to fly halfway around the world to visit family.
Higher salaries often mean more serious teachers, so if you try to negotiate the employer will
A) know you're not desperate (even if you are)
B) know that you know what you're worth
C) know that you've spent time and money getting the qualifications and experience necessary.
Jump Through Hoops
I honestly think that one reason why employers create application forms is to cut down on the number of applicants. It's super easy to shoot off an email, but it takes time and effort to fill in an application form, even if you are copying and pasting from your CV. Employers want someone who will go the extra mile for a good job. FYI: jobs that have you fill out application forms are usually pretty good.
Dress To Impress
If you have an interview, dress well and smile. Employers will decide if they want to hire you or not before you even open your mouth. You have about 10 seconds, yep count 'em, to impress them.

Have Good References and Originals
It goes without saying that you should ask your references' permission before you give their email and phone number out. You should also pick good references. Directors or head teachers are good. Make sure they'll say good things about you as well.
Having your original degrees, certs, and transcript on hand is an absolute necessary with all the fake junk out there. If you don't want to bring your originals, make sure to get copies authenticated or apostillised.