I really think that it's necessary to use "cleats" - that is, glue in square wood backers on all the right angle joints. This will vastly increase the strength of the box. I made my cleats by ripping down a 1x6, so they're 3/4" deep.
You'll also need about 20 #6 x 1" wood screws, wood glue, the correct drill for a clearance hole (bigger than the screw size to drill the top layer of wood) and to do a neat job, also a countersink. As I just used softwood (pine/poplar) I didn't use a pilot hole, I just shot the wood screws into the cleats after gluing.
Gather all the parts:
Lay the cleats onto the bottom plate. Make your marks and decide where you're going to put the screws in. Drill clearance holes and countersinks in the bottom (you don't want the screw heads sticking out the bottom), then glue and screw the cleats down.
Do the same to the front panel, adding the two long cleats along the sides. Be sure to leave room at the bottom for the cleats that you just attached to the bottom.
Drill clearance holes and countersink the triangular side pieces. When you use the side pieces to attach the bottom and front, you need to use a square to make sure that the front is plumb (perpendicular to the base).
What I did was to attach one side to the front first, because that's the most visible so having it nice and flush looks nice. Then I test fit it and determined that in my case it leaned back just a bit (about 1/16") so I glued the bottom cleat, shot the FRONT screw in first, then leaned the front piece out until it was square then shot in the back screw. This leaves a tiny gap under the bottom at the back where it's not visible. You don't want to have the side piece sticking out the bottom as it will make the unit unstable.
Then do the other side the same way and you're done with this part.