I am no expert in the life of Alexander Hamilton, so I can't answer your question with any certainty, but it is highly unlikely he would have known Danish.

While Denmark did purchase St. Croix from the French in 1733, the majority of the non-slaves were of English descent, simply because Denmark-Norway never had any sufficient population surplus to create any substantial Danish population on the Danish West Indies (the other islands being St. John and St. Thomas). Instead Danish authorities relied on getting English (as well as Dutch and French) planter immigrants from other colonies to settle on the island by giving them tax breaks and similar incentives. As followed on St. John and St. Thomas the cultural hegemony was Dutch, but on St. Croix it was English, and Danish culture had a very small role in the daily life on the islands.

In a study of the 300 largest estates on St. Croix in the later 18th century, it was found that the largest group of owners was of Anglo-Saxon descent, comprising 40% of the total. Only 22.7% of the estates were Danish owned, and most of those were owned by Danish colonial officials. There simply didn't exist any significant group of Danish colonial settlers on the island.

This is backed by the fact that the preferred language of St. Croix was English. The most popular newspaper, The Royal Danish American Gazette published between 1770-1802 was mostly written in English and was "essentially English in tone". A travelling theatre group the Leeward Island Company of Comedians visited the island in 1770, performing exclusively in English, among them plays by Shakespeare as well as contemporary plays which were popular in London at the time.

As late as in 1830, when Danish governor von Scholten contemplated instituting a mandatory education for slave children, it was decided to use English as the language of education.

In short, most people used English, or the English based creole (on St. John and St. Thomas the creole was Dutch based), in their everyday life, and most people on the island probably lived their whole life without having to learn any Danish at all.

Source: Slave Society in the Danish West Indies by Neville A.T. Hall, The John Hopkins University Press, 1992. Especially pp. 11-19.