The Ford nine-inch rear-end is, in many ways, the big-block Chevy of third members. It's been the go-to axle for racers of all types since its introduction decades ago. But all nine-inches aren't created equal. Ford built two versions -- a 28-spline unit with smaller axle shafts designed for passenger car use, and a 31-spline unit designed for trucks. The 31-spline retrofit has been popular since there have been 31-spline axles. But don't kid yourself: this one isn't going to come cheap, and there are other options out there for about the same money.
Disassemble the entire axle housing, strip it of every moving component, and have the entire axle hot-tanked. While the 31-spline nine-inch and the 28-spline nine-inch share the same housing and ring-and-pinion set, that's about all they have in common from the factory. Approach this retrofit the way you would an engine build, and start with new, clean parts.
Purchase and install a new 31-spline differential from a name-brand company. Hypothetically, you could just replace the side gears in your existing differential with side gears designed to accept 31-spline axles -- and that is an option, if you're doing a budget upgrade. But the problem with this approach is that, in doing so, you're moving the weak link in the system from the axleshaft to the differential itself. And that's a lot harder and more expensive to replace if you break something. You're far better off installing a stronger diff to go with your thicker axles.
Install new axle bearings. While the primary point behind this upgrade is to utilize a thicker 31-spline axle, a secondary benefit is getting rid of the tapered 28-spline units. Tapered axles are thinner at the differential end than at the wheel-end, and it's impossible to shorten them in order to run a narrower rear-end. The 31-spline axle is the same thickness all the way through; this means you can have it shortened, but it also means that you'll need larger bearings to pass the thicker inner end.
Install the new washers and spacers that come with your 31-spline retrofit kit. There are several little differences in dimensions between factory and aftermarket 31-spline and 28-spline differentials and axles. Any of these can throw off your assembly, and cause your differential to self-destruct.