The Epoch of Reionization (EoR) is the transitional period in the universe's evolution which starts when the first luminous sources begin to ionize the intergalactic medium for the first time since recombination, and ends when the most of the hydrogen is ionized by about a redshift of 6. Observations of the 21 cm emission from hyperfine splitting of the hydrogen atom can carry a wealth of cosmological information from this epoch since the redshifted line can probe the entire volume. The GMRT-EoR experiment is an ongoing effort to make a statistical detection of the power spectrum of 21 cm neutral hydrogen emission due to the patchwork of neutral and ionized regions present during the transition. In this work we detail approximately five years of observations at the GMRT, comprising over 900 hours, and an in-depth analysis of about 50 hours which have lead to the first upper limits on the 21 cm power spectrum in the range 8.1 < z < 9.2. This includes a concentrated radio frequency interference (RFI) mitigation campaign around the GMRT area, a novel method for removing broadband RFI with a singular value decomposition, and calibration with a pulsar as both a phase and polarization calibrator. Preliminary results from 2011 showed a 2 sigma upper limit to the power spectrum of (70 mK)^2. However, we find that foreground removal strategies tend to reduce the cosmological signal signicantly, and modeling this signal loss is crucial for interpretation of power spectrum measurements. Using a simulated signal to estimate the transfer function of the real 21 cm signal through the foreground removal procedure, we are able to find the optimal level of foreground removal and correct for the signal loss. Using this correction, we report a 2 sigma upper limit of (248 mK)^2 at k = 0.5 h Mpc^-1.