Even if the organization is a non-profit one, it has some objectives. So best measures of the performance of a non-profit organization would be their efforts to reach the organizational goals and objectives. Also financial transparency may be used for this purpose.
You can measure non-profit performance according to Input, Output, Outcome or Impact. These measures are further explained in this article (see the framework on p. 64) http://jnbit.org/upload/JNBIT_Sabert_Graham_12(2)_2014.pdf