How can I dog-proof a chain link fence?
The best solution is to add a dig-out guard to your fence or dog run:


Head to the home improvement store and pick up some galvanized sheet metal about 2 feet wide (corrugated or straight is fine; the stuff they use for roofing and sheathing outbuildings is perfect)
With snips, a hacksaw or a Sawzall, cut the metal to workable lengths, maybe 3-4'. If it's corrugated, you want the curves to be perpendicular to the long dimension of each piece, so that when it goes into the ground the curves are vertical. You may, for appearances, paint the "top" edge of the metal a pleasing color of oil-based enamel; then it just looks like edging. But, being galvanized, the metal will be weather-resistant.
Use a shovel (as straight-bladed as possible) to cut down through the topsoil at least a foot and a half (or as deep as possible) along the line of the chain-link fence. You shouldn't have to really dig much; just push down and then wiggle the shovel to widen the cut.
With a block of wood and a rubber mallet, pound the sheet metal down into the cut you made until there's about 6" of the sheet metal remaining above ground. Press any loose earth/grass down firmly against the sheet metal on both sides. Repeat around the entire length of the fence.
With a drill and a metal-cutting bit, drill two holes a couple inches down from the top of the metal and on opposite sides of your chain links, at regular intervals along each piece and where two pieces meet (each set of holes should probably be no more than the width of your dog's shoulders or the maximum diameter of his body).
With linesman's pliers, cut a length of annealed wire, pass it through the holes from the inside of the fenced area, then twist it into place around a chain-link wire on the outside (bored dogs will chew, and you don't want them chewing on metal wire).
Annealed wire will rust over a period of years if given the opportunity (the idea is usually that it's cheap to replace); you can mitigate this with a shot of clear acrylic or polyurethane spray, or you can just keep an eye on the wires and replace them. They do make galvanized wire specifically for use with chain-link, but it's stiffer stuff for a given gauge and so is harder to work with.


You now have an anchored piece of metal that the dog will have to dig more than a foot down to get under (or will hit impenetrable rock before he gets there), that is attached firmly to the chain-link so the dog won't be able to squeeze between them. This should be far more effective than your tent stakes at preventing him getting under the fence.