Lincoln wasn't a third party candidate. He grew up in the Second Party System, which had the Democrats and Whigs face off, but Whiggery is dead by 1860. The Republicans have replaced them as at least the nation's second party by 1856. There are some dead-end Whigs still kicking around, but most ex-Whigs have moved over into the Democracy or the GOP and the remainder are more trying to spoil the election to prevent any of the sectional candidates from winning and thus breaking the Union.

I went into more detail on this about a week ago, if you'd like the who's who and some more of the mechanics. Followup questions are always welcome.

The 1860 election was a four party race. Republican Lincoln won 38.8% of the vote, Northern Democrat Douglas won 29.5%, Southern Democrat Breckinridge got 18.1% and Constitutional Union/Whig Bell 12.6%. The Republican Party came into existence in 1856. They nominated John C, Fremont and he came second in a three way race with 33.1% the winner getting 45.3% and the third place finisher getting 21.5%. So the Republican party had a strong national presence and was the second major party in the country in 1860. If the Democrats had not fractured along sectional lines they probably could have carried the White House but they did not. So while Lincoln was the candidate of a 4 yr old new party and he had the lowest % of popular vote of any president in American history it's inaccurate to call him a third party candidate. 1844, 1848 and 1856 also saw presidents elected with less then half of the popular vote.