1825, "dainty, mincing," from finical "too particular" (1590s), which perhaps is from fine (adj.) + -ical as in cynical, ironical (OED says "ultimate derivation" from the adjective "seems probable"). But finikin (1660s) "dainty, precise in trifles" has been proposed as a source, even though the timing is off. It apparently comes from Dutch; compare Middle Dutch fijnkens (adv.) "precisely, exactly," from fijn, cognate with English fine (adj.).
The -k- between the final -c- and a suffix beginning in -i, -y, or -e is an orthographic rule to mark the pronunciation of -c- as "k" (compare picnicking, trafficking, panicky, shellacked). Related: Finickiness.