The Chromospheric Lyman-Alpha Spectro-Polarimeter (CLASP) sounding rocket experiment, launched in 2015 September, observed the hydrogen Lyα line (121.6 nm) in an unprecedented high temporal cadence of 0.3 s. CLASP performed sit-and-stare observations of the quiet Sun near the limb for 5 minutes with a slit perpendicular to the limb and successfully captured an off-limb spicule evolving along the slit. The Lyα line is well suited for investigating how spicules affect the corona because it is sensitive to higher temperatures than other chromospheric lines, owing to its large optical thickness. We found high-frequency oscillations of the Doppler velocity with periods of 20─50 s and low-frequency oscillation of periods of ∼240 s on the spicule. From a wavelet analysis of the time sequence data of the Doppler velocity, in the early phase of the spicule evolution, we found that waves with a period of ∼30 s and a velocity amplitude of 2─3 km s−1 propagated upward along the spicule with a phase velocity of ∼470 km s−1. In contrast, in the later phase, possible downward and standing waves with smaller velocity amplitudes were also observed. The high-frequency waves observed in the early phase of the spicule evolution would be related with the dynamics and the formation of the spicules. Our analysis enabled us to identify the upward, downward, and standing waves along the spicule and to obtain the velocity amplitude of each wave directly from the Doppler velocity for the first time. We evaluated the energy flux by the upward-propagating waves along the spicule, and discussed the impact to the coronal heating.