Not an especially efficient way to produce it, but could be done in a three stage synthesis as follows:
a) Reaction of acetic acid with thionyl chloride to form acetyl chloride CH3−COCl
b) Reaction of acetyl chloride with ammonia to form acetamide CH3−CO−NH2
c) Reduction of acetamide with lithium aluminium hydride to form ethylamine CH3−CH2−NH2
A more efficient way to produce ethylamine would be to use ethanol as a feedstock and react with ammonia in the presence of sodamide catalyst or something similar.