This is a great question!
The challenge of binary search is that the benefits of binary search come from being able to skip past half the elements at each step in O(1). This guarantees that, since you only do O(lg n) probes, that the runtime is O(lg n). This is why, for example, you can do a fast binary search on an array but not a linked list - in the linked list, finding the halfway point of the elements takes linear time, which dominates the time for the search.
When doing binary search on a file you are in a similar position. Since all the lines in the file might not have the same length, you can't easily jump to the nth line in the file given some number n. Consequently, implementing a good, fast binary search on a file will be a bit tricky. Somehow, you will need to know where each line starts and stops so that you can efficiently jump around in the file.
There are many ways you can do this. First, you could load all the strings from the file into an array, as you've suggested. This takes linear time, but once you have the array of strings in memory all future binary searches will be very fast. The catch is that if you have a very large file, this may take up a lot of memory, and could be prohibitively expansive. Consequently, another alternative might be not to store the actual stings in the array, but rather the offsets into the file at which each string occurs. This would let you do the binary search quickly - you could seek the file to the proper offset when doing a comparison - and for large stings can be much more space-efficient than the above. And, if all the strings are roughly the same length, you could just pad every line to some fixed size to allow for direct computation of the start position of each line.
If you're willing to expend some time implementing more complex solutions, you might want to consider preprocessing the file so that instead of having one string per line, instead you have at the top of the file a list of fixed-width integers containing the offsets of each string in the file. This essentially does the above work, but then stores the result back in the file to make future binary searches much faster. I have some experience with this sort of file structure, and it can be quite fast.
If you're REALLY up for a challenge, you could alternatively store the strings in the file using a B-tree, which would give you incredibly fast lookup times fir each string by minimizing the number of disk reads that you need to do.
Hope this helps!