You need multiple levers, lots of them if necessary. Use plastic levers but I typically carry a couple of types, skinny lightweight and heavier-duty mountain bike levers. I probably own about 4 or 5 lever sets (including an 8-inch long metal monster for downhill mountain bike tires).
First, inflate the tube slighly just enough so it stays out of the way of the tire bead. To put a tire on, start with your fingers (if possible). When the fit starts to get too tight for fingers, slip levers at intervals under the unseated bead. Start levering the bead onto the rim with each of your levers. It will get tougher and tougher to lever the bead up over the edge of the rim. This is where you might need heavy-duty levers (I have snapped lighweight levers at this point). Keep working around the bead. Eventually, you might find you can lever the bead onto the rim sidewall, but not fitting properly over the rim-edge. If you can't slip a lever under the bead while it is stuck at rim-level, you need to go back to using your fingers. Start at edge, just where the tire bead come off the rim and goes in a straight line across the rim sidewall. Put your fingers of the rim ad push outward with your thumbs at the edge of the rim. Keep moving across. If you can't push on that one side any more (the bead has become too tight), start from the other side. Eventually, it will snap on, or you will have a very small piece of the bead hanging off the rim that won't budge. In this case, you might need a blunt metal object (I use the edge of a metal lever) to push the bead up over the edge, being careful not to scratch the sidewall. Generally, I've never had to go further than this. If you can't get that last bit up, I would suggest you leave it for a while (say overnight or longer) to stretch in that position.
If it seems hopeless or impossible, you probably need to shop for new tires. If your tube keeps getting pinched by the tire bead, you probably should try a smaller tube size. Hope this helps!