Is it that you actually want to climb the Matterhorn? You can do this from either Breuil-Cervinia/Italy or from Zermatt/Switzerland.
If you just want to get near it and see how it looks like, the obvious is to travel by train from Basel to Zermatt. The ride takes between 3 1/4 and 3 3/4 hours. For a personalized schedule as well as ticket prices, click here. Don't rent a car, you can't drive up to Zermatt, you would have to leave it in Täsch and pay a lot for the parking there, then still change onto the train.
Zermatt has a campground, click here to see it. It actually opens June 1st as right now, there is still snow up there. Forget about wild camping - again, because there is snow up there. Wild camping up in the mountains is however theoretically possible in Summer time - if you can stand the cold during the nights and if you're willing to schlepp all the material such a thing requires.
As on how to go from Zermatt to Salzburg, use the link above. It will be a full day's ride.