Outdoing rhymes with:
dewing, doing, overdoing, redoing, subduing, undoing, accruing, blueing, bluing, booing, brewing, canoeing, chewing, cooing, cuing, debuting, dewing, doing, eschewing, ewing, hewing, lewing, misconstruing, nonaccruing, overdoing, pursuing, queuing, redoing, renewing, reviewing, screwing, skewing, spewing, stewing, subduing, suing, tattooing, undoing, viewing, wooing
Outdoing sounds like:
o'donoghue, odoms, odonoghue, oetting, othman's, otten's, ottens, otting, outing, outings, outwitting