Is it possible to get the job if I am not a native but I have very good English background and speaking?
It is possible to get English teaching jobs for non-native speakers with very good English but usually these are lower paid or less attractive jobs. Also, although it is illegal many smaller schools try to recruit good looking younger white teachers. Sometimes they are also looking for a teacher of a specific sex, either male or female. They will never admit to this.
There is a shortable of able teachers - particularly for children, so if you are an experienced and talented kids' teacher don't be afraid to say so.