Normally I'd say just put the seal in and you'll be fine, but for some reason, the passenger-side axle seal gave me trouble the first time I did it too. I have no idea how I did this, but the first time I did one, I pushed the old one into the transmission while installing the new one in its place. It is the strangest thing. It worked, but I eventually took both seals out, and installed a 3rd.
There really is not much to installing one, just make sure you drain the fluid, pull the axle out, remove the old seal, and then tap in the new seal. Be careful to tap the new seal in by the outer edge only, where the metal ring is. And tap it in a little bit at a time, taking care to travel around the seal so it goes in small amounts each bit as you go full-circle multiple times around the seal.
I guess the other thing to do is make sure your transmission doesn't have any scrapes around the seal surface.