We investigate the hydrogen chemisorption mechanism on both pristine and Li-doped (5,5) single-wall carbon nanotubes (SWNTs). The chemisorption of $H_2$ molecule is found to be suppressed due to very high energy barriers (~3 eV), while H atoms are readily chemisorbed with no energy barrier on pristine SWNTs. When $H_2$ is applied to the Li-doped SWNT, the hydrogen chemisorption is facilitated by a successive LiH chemisorption and the resulting barrier reduction for the $H_2$ chemisorption. Our calculations show that the hydrogen chemisorption is more probable in Li-doped carbon nanotubes, compared with pristine carbon nanotubes, for a hydrogen storage.