The majority of EFL teachers in Greece are from Britain and Ireland. As Greece is a member of the European Union, there are no barriers to acquiring visas or permits to live and work in the country. For non-EU citizens, the situation is far more complicated.
If you are a national of the US, Canada, Australia or New Zealand you can enter the country for up to 90 days as tourist, without needing a visa. This would allow you to search for a job but you will not legally be able to engage in paid employment. Some schools may be willing to hire teachers temporarily on this basis but benefits are likely to be low and you run the risk of being deported and banned from re-entering the country for up to ten years.
To legally obtain work and residence permits, non-EU citizens must first obtain a formal job offer from a prospective employer. It is unlikely that any employer will offer a position to a stranger based in another country because they are required to put up a deposit for the application process and will have to jump through a succession of bureaucratic loops. The only way a non-EU citizen will likely receive a formal job offer is if they have friends or relatives based in Greece, or they have traveled to Greece beforehand to secure a position.
Assuming you have traveled to Greece and have been offered a job, you will then need to return home to continue the application process. First, you need to apply for a work permit. Before you can do this, your prospective employer must submit all the relevant paperwork to the OAED, a bureaucratic manpower agency. In order for the application to be accepted, they are required to prove that a Greek or EU national in any of 27 member states cannot be found to do this job instead of you. Perhaps, the biggest obstacle to non-EU citizens obtaining a job offer, though, is that a deposit is required for the application process to begin, equivalent to several months salary. In some cases, the employer will expect you to put up the deposit yourself.
If the application is successful, and in many cases it is not, a formal offer of employment and work contract will be forwarded to the Greek Consulate in your home country. You will then be invited to the consulate for an interview, which will hopefully result in a visa being issued. This process will take a minimum of 2 months but has been known to take up to a year!
Once you have been issued with a work visa, you may travel to Greece and apply for work and residence permits, a tax number and register for national insurance. All of this paperwork must be completed within 30 days of your arrival, otherwise you will be forced to leave the country and begin the application process again.
Needless, to say, there are far easier options in other countries for non-EU citizens who wish to teach in Europe.
A general overview of what to expect teaching English in Greece
Detailed living costs for teaching English in Greece
TESOL Greece An independent, non-profit association of EFL teachers in Greece.
TESOL MACEDONIA-THRACE An independent, non-profit association of EFL teachers working in northern Greece.
Greek National Tourism Association (Travel guide)