The northern border (along with the western border) was ill-defined in the original treaty, in a large part because both areas were almost entirely unknown to European sources. Jefferson primarily viewed the purchase as a way of acquiring sovereignty over New Orleans, and with it, the Mississippi. Everything else was, essentially, gravy. You can get an idea for how little clarity there was on this topic from the 1803 treaty document. In particular, this paragraph is the closest the negotiators got to geographically defining the nature of the territory:

His Catholic Majesty promises and engages on his part to cede to the French Republic six months after the full and entire execution of the conditions and Stipulations herein relative to his Royal Highness the Duke of Parma, the Colony or Province of Louisiana with the Same extent that it now has in the hand of Spain, & that it had when France possessed it; and Such as it Should be after the Treaties subsequently entered into between Spain and other States.

The fact that they spend multiple articles describing the disposition of public buildings, fortifications, and river navigability, and almost nothing about the wilderness frontier of the purchase, tells you almost everything you need to know of what they were primarily thinking about.

That's not to say it didn't matter, though, hence the organization of expeditions specifically tasked with shedding some light onto the extent and nature of the territory (the Corps of Discovery under Lewis and Clark being the most famous of these). The US interpretation of the treaty was that it represented the entire drainage basin of the Mississippi lying west of the river. In modern terms, this would encompass an area which carves off about half of North Dakota, roughly two-thirds of Minnesota, most of Montana, and a sliver of what is now Alberta and Saskatchewan. You can get a general idea of what this would have meant by looking at the hydrological basins of North America. Wikipedia has a nice high-level one, with the green line and purple lines roughly denoting the northern reach of the territory.

Of course, none of this was known in such detail (or anything close to it) at the time. And even the interpretation of the purchase as denoting the entire Mississippi basin was bitterly contested by Spain and England. It would need another 16 years and two additional treaties before the matter was essentially settled, and (among other things) the northern boundary of what we now know as Montana, Minnesota, and the Dakotas was fixed along the line that we now recognize.