The ability to speak some German will increase your chances but is certainly not a deal breaker. Also if you have experience teaching Business English that will help your chances as well. Any additional skills help here as Austrian language schools seek to differentiate themselves by offering more than just traditional general English courses. They also like to build long-term relationships with their teachers so if you plan on staying longer this helps.
In private language schools most teachers work on a freelance basis so contact the individual language schools for information about their employment criteria and the type of contract they offer.
UK and Irish citizens will find it easier to find employment due to their right to live and work in any European Union country. Americans and other native English speakers will have a harder time as getting a work or residence permit is notoriously time consuming. If you are a non-EU citizen you should contact your embassy to see what the specific requirements are for you to work in Austria.
A work permit to teach or tutor English is not required in Austria if you are self-employed. There is however a license (Gewerbeschein) required if you wish to form classes of 5 students or more.
If you are freelancing and have a work contract (Werkvertrag) with a firm, however, it is important to distinguish whether you are truly self-employed or whether you are an employee of the firm. The most important issue legally is how dependent you are on the firm. Make sure you are aware of what is involved with this.
Teachers of English in Austria - non-profit organization for TEFL teachers in Austria