stalk (n.) Look up stalk at Dictionary.com
"stem of a plant," early 14c., probably a diminutive (with -k suffix) of stale "one of the uprights of a ladder, handle, stalk," from Old English stalu "wooden part" (of a tool or instrument), from Proto-Germanic *stalla- (cognates: Old English steala "stalk, support," steall "place"), from PIE *stol-no-, suffixed form of *stol-, variant of root *stel- "to put, stand" (see stall (n.1)). Of similar structures in animals from 1826.
stalk (v.1) Look up stalk at Dictionary.com
"pursue stealthily," Old English -stealcian, as in bestealcian "to steal along, walk warily," from Proto-Germanic *stalkon, frequentative of PIE *stel-, possibly a variant of *ster- (3) "to rob, steal" (see steal (v.)). Compare hark/hear, talk/tell). In another view the Old English word might be from a sense of stalk (v.1), influenced by stalk (n.). Meaning "harass obsessively" first recorded 1991. Related: Stalked; stalking.

A stalking-horse in literal use was a horse draped in trappings and trained to allow a fowler to conceal himself behind it to get within range of the game; figurative sense of "person who participates in a proceeding to disguise its real purpose" is recorded from 1610s.
stalk (v.2) Look up stalk at Dictionary.com
"walk haughtily" (nearly the opposite meaning of stalk (v.1)), 1520s, perhaps from stalk (n.) with a notion of "long, awkward strides," or from Old English stealcung "a stalking, act of going stealthily," related to stealc "steep, lofty."