There is a high demand for English language teachers in Hungary and its not too difficult to find work if you have TEFL qualifications.
Many travellers don't bother with Hungarian bureaucracy and work under the table, this is pretty easy especially if you're giving private English lessons.
If you want to get all the legal paperwork you'll need to get your employer to get the ball rolling and organise a work permit. Once you've got a work permit, you will need to get a residence permit from the KEOKH office. Agencies such as Settlers Hungary can get all this paperwork sorted out for you.
KEOKH
VI, Városligeti fasor 46/48 Metro Hõsök tere Tel 118 0800