You cannot use permanent magnets alone (it's a physics impossibility, a consequence of Earnshaw's Theorem). You need to lawyer your way out of it, which you can do either by removing "permanent" (i.e. employ active electromagnets also; this is what is commonly done with magnetic suspensions for lamps etc.) or by removing "alone".
it has been pointed out to me that Earnshaw's Theorem only holds with ferro- and paramagnetic sources. The addition of diamagnetic materials does allow stable configurations with permanent magnets alone. Werner Braunbeck's trick, though, only holds with diamagnetic materials, and those have a strength that's whole orders of magnitude below high-flux rare earth ferromagnets. So, unless the wheel is large enough to support a half ton of diamagnetic bismuth shield, I'm afraid passive levitation isn't a thing for bikes.
Earnshaw's Theorem actually applies to a whole class of forces, so you can use no combination of those to wiggle out of "alone". No electrostatic fields, no gravitational fields. You're pretty much left with superconducting pinch-effects and, again, mechanical bearings (you can replace a mechanical bearings assembly with permanent magnets as long as you keep at least one bearing).
You could have a spokeless, hubless wheel where the weight is borne by magnets between an inner rim and the tyre, with mechanical bearings to keep the two rims aligned (the bearings would only be subject to lateral forces). This allows the magnets to spread out the weight and thus exert a minimum force.
And at this point you need to keep grit out of the space between the magnetic poles. You need a seal, and that's going to need to be mechanical, and problems with friction will resurface. Also, the magnets will be heavy. A toroidal N41 magnet, circularly magnetized (north pole in the hole, south on the periphery), 4cm wide, 2cm thick and 5cm of inner radius, weighs about six kilograms; you need another wider segment to stay above, and that will probably weigh two to three kilos; this will get you a repulsion of around 200 kg, i.e., unless you thrust 2000 newton of force downwards, the magnets won't touch. One advantage is that it will double as suspension. A big disadvantage is that anything magnetic will be attracted to the rims with great force (you can harm yourself pretty severely with those kind of babies). And finally, you'll have a rotating thingy inside a very powerful magnetic field. Chances are that you'll be getting fierce eddy currents, and a considerable magnetic drag.