Welcome to the hobby!
You can totally use a 5 gallon kit for 1 gallon batches. It's a bit overkill size wise but that's not a big deal. You may find you want a smaller fermenter but these can be had fairly inexpensively in the form of 2 gallon food grade plastic buckets.
I suspect, if you enjoy beer and you discover that you enjoy making beer 1 gallon is going to disappear pretty quickly. Remember it takes about 3 to 4 weeks to make a batch of beer whether it is 1 gallon or 10. Lots of folks on here brew 2 or 3 gallon batches because 5 is just too much for them or becuase they like lots of variety and a 5 gallon brew kit will work just fine for that.
Loking at Northern Brewer, it looks like they include a recipe kit with the deluxe brewery kit. Which one did you choose?
If you want to let us know some commercial beers you enjoy we will also enthusiastically give you style and even specific recipe recommendations as you progress.
Again, Welcome! and ask any questions you might have freely. We are happy to help!
Thanks very much for your reply. your advice is relieving. I actually chose the Caribou Slobber because I didn't love any of the other recipes. I know that the CS recipe calls for a secondary fermentation, for which it will be useful to have the Deluxe Kit. My very favorite types of beer are generally Bourbon Barrel Aged. Have you ever a batch with a bourbon addition? Thanks again!