The best method of building a door, if you don't have a router table and the special bits, is to miter the corners then drill and counter sink for the screw heads. Glue each joint before you put the screws in and you will have strong joinery. A diagonal from top to bottom, mitered in then glued and screwed will make sure the door frame remains square.
The brackets you are talking about also work fine and are much easier to work with especially if you don't have much carpentry experience or if your in a hurry.
Be sure you lay out your frame boards square (3-4-5 method) works very well for making sure you are square. again you would want to put in a diagonal but with the brackets you just lay the diagonal where it will fit then scribe from the underside for your cut lines, once that's done, drill glue and screw.
I like to use Decking screws for building doors, they are stronger than cabinet or drywall screws and not much more cost wise. They come in up to 4" lengths at half inch increments from 2" long the short ones are 1 7/8" long. I have found that having 3 1/2" and 4" makes putting together a really solid door easier. For the brackets, you can get away with using 2" or 2 1/2" screws.
Right now we have a solid door installed but at some point in the future My wife wants me to make her a Dutch double door for the coop. If we are able at that time I will put up the photos of how to do the build.