The first step is to make sure the top edge of the box is straight and square. I do this by planing around all of the edges with my block plane. I use a piece of plywood after to see if there are any gaps. 

Next I find the center of the box and mark it

The lockset I got was really inexpensive. As such it's a huge pain to install. There are no alignment tabs to speak of. To find the position of the key, I used my calipers to measure down from the top of the lock to the center of the key hole. I transferred that measurement to the outside of the box

I measured the diameter of the key and drilled a hole slightly larger. In this case it was 5/16"

I then measured the width of the lock itself, divided it in half, and marked a line on each side of my center. I used my marking gauge to set the baseline. Because the box is already assembled, it's going to be difficult cut this mortise. I went as deep as the gauge would go.

I then clamped the box to the bench so I can begin cutting the mortise

This is called a woodpecker saw. It as a special tooth on the front that will let me start a cut in the middle of a board instead of just on an edge. I used this to saw down to my baselines on the edges 

I then began clearing some waste with my mortise chisel .I had to be careful because the scribe in the back wasn't very deep, and all of my chisels are too tall to fit in the box. What I did was cut to the scribe line depth to make a ledge. 

Not wanting to risk splitting out the baseline with a chisel, I switched to my router plane to clean it up

After a few passes, I would use a right and left handed single bevel marking knife to deepen the baseline. The wide bevel gets a good registration against the ledge, and I was able to get right into the corners. A spear point wasn't working as well because there was not quite enough registration. If you care careful, you can make it work. Also, butt hinges would have been a really good idea (if I had any)

I continued with the router planes and chisels until I reached all of my marked lines. It worked and the lock fits really snug

Now to cut the shallow mortise for the top. The ends are rounded and have a 3/8? radius.  I put the lock in the mortise and started the screws. 

Using the screw holes as pilots, I used a 3/8" forstner bit and drilled to the depth of the lock plate. I set my calipers again to be sure I didn't go to far. After the holes were drilled I used chisels and my router plane to clean up the rest