The Lick Planet Search program began in 1987 when the first spectrum of τ Ceti was taken with an iodine cell and the Hamilton Spectrograph. Upgrades to the instrument improved the Doppler precision from about 10 m/s in 1992 to about 3 m/s in 1995. The project detected dozens of exoplanets with orbital periods ranging from a few days to several years. The Lick survey identified the first planet in an eccentric orbit (70 Virginis) and the first multi-planet system around a normal main sequence star (Upsilon Andromedae). These discoveries advanced our understanding of planet formation and orbital migration. Data from this project helped to quantify a correlation between host star metallicity and the occurrence rate of gas giant planets. The program also served as a test bed for innovation with testing of a tip-tilt system at the Coudé focus and fiber scrambler designs to stabilize illumination of the spectrometer optics. The Lick Planet Search with the Hamilton Spectrograph effectively ended when a heater malfunction compromised the integrity of the iodine cell. Here, we present more than 14,000 velocities for 386 stars that were surveyed between 1987 and 2011.