If Mexico was the nation that conquered Hawai'i, instead of the US, what would the colony be like? Culture, sociopolitical structure, and so on.
Which regime/administration would be most able to do this?
And how long would it take for Hawai'i to be made a state?
-- Due to the high degree of improbability, your
if is a
big one. It would take a series of PODs from early in Mexico's history to raise the rabble of butterflies necessary to bring this to pass. Not only was Mexico too politically unstable and militarily inferior to pull it off, Great Britain and France, two global powerhouses, had a military presence in the islands early in the 19th century and would head-off any Mexican attempt to conquer Hawai'i at the pass, as would the US.
-- No OTL regime/administration would be able to do it. That said, "most able" among the OTL leaders might possibly be José de la Cruz Porfirio Díaz Mori, and only because he ruled the longest.
-- As it would take a series of PODs from early in Mexico's history to raise the rabble of butterflies necessary to put Mexico in the position of being able to take Hawai'i for itself, the only answer to the question of
when would Hawai'i become a Mexican State is, "it depends on the POD".