If you already know that water won't run uphill and how to handle a drill, you can do this. The most ingenious part of the design is the hose-to-barrel connection. Since the attached hose will frequently be tugged during normal use, it is important to use a mechanical connection rather than a glued connection.
This design uses a simple garden hose washer, standard garden hose parts, and a special adapter, that's not really all that special. It's expandable and useful for more than just capturing rain water.
Step 1: Got Parts?
Also, while in town, pick up a garden hose valve, garden hose washer, and a MHT to FPT 3/4" plastic adapter. What? MHT = Male Hose Thread. FPT = Female Pipe Thread. Plastic = plastic. You'll know it when you see it. Less than $10 total.
I had some old parts handy to draw from.
The gutter modification comes later.
Step 2: Tools Needed
To modify a gutter downspout, you will also need a heavy duty pair of scissors (or sheet metal snips), about 10 small sheet metal screws, and a small drill bit to pre-drill for the screws. A pop-rivetor would also work. The special pliers shown in the photo are for "shrinking" sheet metal. Optional.