I am not the first to make a chair out of a shopping cart; google it and you can see a lot of possibilities. So, I approached the problem with a mind towards simplicity, making just a few precise interventions. It took me less than two hours to make. Tools are minimal: an angle grinder with a metal cut-off wheel, a dremel or equivalent, or even bolt cutters, two pieces of scrap wood, and two clamps. A drill could come in handy depending on the condition of your shopping cart.
The chair is really comfortable: the front wheels elevate the front of the seat and create a nice recline. The seat is deep and roomy, and the arms are wide, at a low height. Since shopping carts are made to sit out in the weather, this could be an indoor/outdoor chair. With the handle intact and the wheels in the front, it is easy to lift a little and wheel around. On my flickr page here, you can see another shopping cart chair a buddy and I made, with a shopping cart chassis and some oven grates.
I found mine in an alley, missing its front wheels. I think, as an experienced dumpster diver, you have to make a judgement call on things. I would recommend taking a cart only if it is nowhere near a relevant supermarket or other store and/or it is somehow damaged but salvageable for our purposes. Other possible places to get carts are the local/county dump, private metal scrap yards, craigslist, stores going out of business, or asking at a store what they do with damaged carts. Shopping carts are apparently quite expensive for retailers to purchase, and they expect them to last a long time. Don't steal carts.
Step 1: Off With the Wheels
Step 2: Seat
Step 3: Arms!
Use the grinder to snip the two welds that fasten the basket sides to the back frame, cutting only down to where the wood has been clamped. All carts are different, so look and cut carefully. You don't want the sides completely detached from the back, only halfway detached, to maintain some rigidity.
Then just bend the sides of the basket over. The clamped wood just provides some resistance at the bend, so as to make a clean crease.
Step 4: Reattaching Wheels
This step is optional.