It’s not an official contraction, but merely an easier way for native speakers to pronounce the -ing ending in rapid speech. This is a process that can apply to all verbs, not just “to do”. In formal speech, we prefer to use the full pronunciation of -ing. Note that while we write it as “doin’”, it’s not really a matter of dropping the “g” — in fact, there is never any /g/ sound at all (“g” represents /ŋ/, as in Hanyu pinyin). Note the reduction of /ɪ/ to the schwa
/ə/ (which may even be dropped entirely):
doing: /dʊwɪŋ/
doin’: /dʊw(ə)n/