A number of international organizations have offices in Bishkek, however most employees are recruited from abroad. If you speak Russian, there might be occasional opportunities to find temporary or long-term work. There are also a number of English language schools that will employ native English speakers.
Bishkek is a cheap place to learn Russian or Kyrgyz. A private 1 1/2 hour lesson with a native Russian speaker should cost between $5-7. Courses are also available at the American University of Central Asia (http://www.auca.kg/) and the Kyrgyz-Russian-Slavic University (http://www.krsu.edu.kg/).
There is also a private school that caters to individual learning: The London School in Bishkek (http://www.tlsbi.com). This school offers Russian and Kyrgyz to anyone at anytime of the year for as little as 120 soms/hr. During the warmer months they are often full so book in advance.