You have to get an E-2 visa to work as an esl teacher in Korea and its process is as follows.
『 Basic qualification required for working as an ESL teacher in Korea. :
  ① You must be a native English speaker.(English must be your first language.)
  ② You must have acquired a bachelor's degree or higher from that country that uses English as its
      first language.
(Ex. of the mentioned country: USA, Canada, UK, Australia, New Zealand, etc) 』

1. You must send us the following six(6) kinds of documents to be
  submitted to our immigration office.『 You must send them by DHL or FedEx or other
    secure couriers, and then email us the 'tracking number' for your DHL or FedEx package. 』

  (1) A photocopy of your passport (picture page).
    『 The validity of your passport must not expire within six(6) months. 』

  (2) Self Health Statement.
    - Based on the government employee regulation accompanying paper.
      Philopon, Cocaine, Opium, Hemp are the essential points for the drug test, HIV test is also an
      essential point.
     (Self Health Statement Form: Please download it at 'Documents Download' on the menu.)

  (3) Degree Certificates
    - Got Apostille or verified from the Korean embassy in foreigner's home country‘confirmation certificate
      from the university’ or ‘graduation certificate including degree achievementfacts’ or ‘diploma
      or degree copy’
    * If a foreigner hands in a degree certificate for the visa issuance application, and comes to Korea
      after getting the conversation teacher(E-2) visa, and if the document is made as data in the immi-
      gration system, the degree certificate is able to be omitted when the foreigner wants to move to
      another workplace through a new visa issuance application.
    * Submitting the sealed transcript was omitted.*Your full name both of the degree certificate and the
      Apostille must be same as that of your passport.
    → http://www.apostilleinfo.com

  (4) Issuance of proof of (lack of) criminal records
    - Criminal Record Check with Apostille is valid for only 6 months from the date of its issue.
    - A criminal record check which covers your whole country, and also from your country's
      government.
    - The records must be affixed with the relevant nation's Apostille.
    ※ According to the Korean immigration law a criminal background check from the Korean Ambassa-
       dor of your country will not be accepted.
    ※ Nationals of countries that have not signed the Apostille treaty, (China, etc) must get their records
       verified by their local Korean consulate and those already in Korea who were issued their criminal
       records from their consulates in Korea have no need for an Apostille
    ※ In case applicant stay in Korea and is sent Criminal Background Check by mail, Or one's entry to
       Korea is imminent, the document notarized by one's embassy in Korea may be exceptionally
       acceptable.
    ※ Immigration office will not recognize the criminal records issued by online website or E-mail
       notarized by consulate's office in Korea as a valid criminal records.
    - If a foreigner hands in a criminal record check which covers the whole home country's information
     for the visa issuance application, and comes to Korea after getting the conversation teacher(E-2)
     visa the criminal record check document is able to be omitted when the foreigner wants to move to
     another workplace through a new visa issuance application.
     (This includes the situation such as, visa issuance application made after the foreigner left Korea.
     But, if the time period from leaving date up to entrance date with the new E-2 visa is more than
     3 months criminal record check is needed when alien registration application is made.)

* NOTE
  ① Consulates located in Korea cannot issue proof of criminal records to individuals currently.
  ② The Korean government is asking for a official criminal record check from your own country's
    government, but information of how to get the criminal record check document should be asked to
    your own country.
※ If obtained through consulates in Korea, please refer to the list below or the local webpage.
  - U.S. : FBI CJIS DIVISION(Federal bureau of investigations),
            http://www.fbi.gov/hq/cjisd/cjis.htm
  - Canada: Royal Canadian Mounted Police, http://www.rcmp.ca
  - New Zealand : Ministry of Justice website, http://www.justice.govt.nz/privacy/

  (5) Two(2)pieces of passport size (3.5㎝ x 4.5㎝)color photos.
     (Photos must have been taken within the past 6 months.)

  (6) Resume with correct address and contact information

2. Your employer submits 'Application for Confirmation of Visa Issuance'
  to our Immigration Office.〔 That application must include all of the documents men-
   tioned in the item No.1 above. 〕

3. Our immigration office provides your employer with the visa issuance
  number.〔 It takes about two weeks to get the visa issuance number. 〕

4. VivaCom informs you of the visa issuance number by email
  and/or phone.

5. You have an interview with Korean Consulate to get the E-2 visa.
〔 Process:
  (1) You submit the following two(2) kinds of documents to Korean consulate.
    ① Application of Visa Issuance(with visa issuance number of item No. 4 above).
    ② Passport.
  (2)According to the standard interview manual, conduct a face-to-face interview. However, if
    circumstances were such that making a visit is not possible or plausible, then it may be conducted
    over video conference. However, you must confirm preliminarily via phone the process and
    documents to get the visa because they are changeable any time.〕

@Most of the above informations were quoted straight from the website of our immigration office.