We present an analysis of high-precision pulsar timing data taken as part of the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) project. We have observed 17 pulsars for a span of roughly five years using the Green Bank and Arecibo radio telescopes. We analyze these data using standard pulsar timing models, with the addition of time-variable dispersion measure and frequency-variable pulse shape terms. Sub-microsecond timing residuals are obtained in nearly all cases, and the best rms timing residuals in this set are ~30-50 ns. We present methods for analyzing post-fit timing residuals for the presence of a gravitational wave signal with a specified spectral shape. These optimally take into account the timing fluctuation power removed by the model fit, and can be applied to either data from a single pulsar, or to a set of pulsars to detect a correlated signal. We apply these methods to our data set to set an upper limit on the strength of the nHz-frequency stochastic supermassive black hole gravitational wave background of h sub(c)(1 yr super(-1)) < 7 x 10 super(-15) (95%). This result is dominated by the timing of the two best pulsars in the set, PSRs J1713+0747 and J1909-3744.