This means underhand or unscrupulous behaviour or trickery. There is doubtless no more activity of this kind in the USA than anywhere else, but it was in that country that the word was coined, sometime around the middle of the nineteenth century. The first recorded instance I know of is this:
If the above be true, and we are told that ’tis so, we see nothing else in the new move but a bit of political skulduggery for the benefit of black republicanism; but it will not redound to their benefit — it will return to plague the inventors of the plot.
The Burlington Daily Hawk-Eye (Iowa), 28 Aug. 1858.
The word took some while to catch on. One early sighting is in this splendid political exchange, which I have gleaned from the Official Report of the Proceedings and Debates of the Third Constitutional Convention of Ohio, 1873-1874:
Mr. WEST. It is urged upon the assumption that there has been what some gentlemen here have characterized as “smouzling”.
Mr. HOADLY. What is that?
Mr. WEST. Skulduggery.
Mr. HOADLY. Well, what does that mean?
Mr. WEST. I do not know what it means, but that is what I heard talked about here.
We are hardly the wiser as a result, except that the members of the Ohio legislature didn’t know what skulduggery meant, except that it was something that could be considered vaguely disreputable. Skulduggery seems to have caught on in the US quite slowly, and was imported into Britain only much later.
Though we now know pretty well what it means, its origin remains unclear. Experts think that it is probably Scots, most likely from sculduddery. In the eighteenth century, sculduddery meant fornication, adultery, unchastity. In the nineteenth century it seems to have shifted to a sense of obscenity and indecency in language. Later it was respelled and changed again to its current meaning. Unfortunately, nobody seems to have the slightest idea where sculduddery comes from, so here the trail runs into the sand.