We present the target selection , observations , and data reduction and analysis process for a program aimed at discovering numerous , weak ( equivalent width \leq 100  mà ) Ly \alpha absorption lines in the local Universe ( 0.003 \leq z \leq 0.069 ) . The purpose of this program is to study the physical conditions of the local intergalactic medium , including absorber distributions in Doppler width and H i column density , redshift evolution of absorber number density , line-of-sight two-point correlation function , and the baryonic content and metallicity . By making use of large-angle , nearby galaxy redshift surveys , we will investigate the relationship ( if any ) between these Ly \alpha absorbers and galaxies , superclusters and voids . In Paper I , we present high resolution ( \sim 19  km s ^ { -1 } ) spectroscopic observations of 15 very bright ( V \leq 14.5 ) AGN targets made with the Goddard High Resolution Spectrograph ( GHRS ) aboard the Hubble Space Telescope ( HST ) . We find 81 definite ( \geq 4 \sigma ) and 30 possible ( 3 - 4 \sigma ) Ly \alpha absorption lines in these spectra , which probe a total pathlength of 116,000 km s ^ { -1 }  ( \Delta z \sim 0.4 ) at very low redshift ( z \leq 0.069 ) and column density ( 12.5 \leq \log { N _ { HI } } \leq 14.5 ) . We found numerous metal lines arising in the Milky Way halo , including absorption from high velocity clouds in 10 of 15 sightlines and numerous absorptions intrinsic to the target AGN . Here , we describe the details of the target selection , HST observations , and spectral reduction and analysis . We present reduced spectra , absorption line lists , “ pie diagrams ” showing the known galaxy distributions in the direction of each target , and nearest galaxy tables for each absorber . In Papers II and III , we use the data presented here to determine the basic physical characteristics of the low- z  Ly \alpha forest and to investigate the relationship of the absorbers to the local galaxy distribution .