It's a simple question with a not so simple answer.
One way to do it would be to make the 3rd party as a whole viable and a major player in the game.
It'll probably take a major voting bloc of one of the big parties to break off and either join a 3rd party or start their own in order to be taken as serious.
Even if they do this, though, more then likely they'll just end up joining up together again with one of the two big parties, because if for example the evangelicals broke off from the Republican party, and split that vote in every district so that the Democrats won all the seats, they would be worse off then they were before, where they at least got some representation. Thus, they rejoin.
The other way would be on an individual level, the Ross Perot way. If you have tons and tons of money, and then get exposure (such as the debates where the perception was that Perot won), you can show people that you are a viable candidate and poll numbers will support this claim. If Perot hadn't have backed out of the election for a couple months, he would have been a strong competitor.
The thing is, though, people won't be convinced that whatever party this candidate belongs to is going to be worth voting for every single time. They're voting for the candidate, not the party.