First, let's agree that we would prefer a representative democracy where the winning candidate is supported by the majority of the voters (even if their support is "I hate the other more"). If you don't agree then please go away, I've got nothing to say to you and I'm not interested in what you've got to say.
Second, since we don't have job-sharing for elected office, it's a simple truth that if more than two people are on a ballot then the winner might not reflect the will of the majority of the voters.
As an example, Bill Clinton won the presidency in 1992 with only 43% of the popular vote. George Bush (the first) and Ross Perot were supported by 56%. However, the Electoral College is an amplifier because the most states are "winner-take-all," awarding all the electoral votes the candidate with 50.00000001% of the popular vote. Clinton won 69% of the Electoral College and claimed a landslide while the GOP simultaneously said the popular vote showed he wasn't supported by the American people. You might recall a bit of divisiveness at the time over whether Clinton had "a mandate." Would Perot voters have preferred Bush, Clinton, or stayed home if given only a binary choice? We won't ever know. In any case, we effectively elected a president who did not unequivocally have the majority of the voters behind him (we did it again in 2000, but that was more of an appointment than an election, which is another story).
If the president doesn't have the explicit support of the majority of the voters can you really call us a representative democracy? Seems to me that a system with a leader who does not reflect the will of the majority is typically known as a "People's Republic".
A starting point would be a constitutional amendment that requires states to apportion electors according to the percentage of vote rather than winner-take-all. Unfortunately, if you leave it there you run into a problem: if no one gets 270 electoral votes the US Constitution requires the House of Representatives to decide the presidency*. This approach made sense in the 18th and 19th centuries when transportation and communication were slow, but is absurd today. Thus, any change to the Electoral College should also require a run-off election between the top two candidates when no one has a majority.
Of course, it might be simpler just to write an amendment that abolishes the Electoral College and require all presidential elections to be two-step: first, a vote with candidates from all parties on the ballot, followed by a run-off between the top two.
A run-off is the only practical way for a third-party candidate to actually get elected to office. A third-party candidate is unlikely get enough votes to beat both major party candidates because that requires appealing to disgruntled voters of both parties**. However, it might be possible for a third party candidate to beat one of the major party candidates. Imagine if the GOP establishment today got totally behind the Libertarian Gary Johnson with money, advertising, and endorsements. It is quite possible that Johnson could beat Trump. But in November's vote, there is essentially zero chance that Johnson will draw enough Democrats to beat both Trump and Clinton in a 3-way race, and even less of a chance in the 4-way race where some Bernie supporters will choose Jill Stein. But if Johnson beat Trump and we had a run-off of Clinton vs. Johnson, who knows where the Bernie supporters and "hate Clinton but hate Trump more" voters would fall?
The Electoral College and winner-take-all system ensures third party candidate can never be more than a spoiler to the dominant party that is most similar***.
Fix the system before you ask me to waste my vote.****
* In 1992, Clinton would have only won 238 electoral votes if they were proportional, but the House of Representatives was Democrat so he would have been President anyway. The requirement to go to the House for resolution is also (arguably) why Scalia et al on the Supreme Court interfered in the 2000 election - strictly following both the constitutions of both the US and Florida would likely have made it impossible for Florida to have electors designated in time for the vote of the Electoral College. It isn't unconstitutional for a state to screw up, it would have simply meant that neither Bush (II) nor Gore got 270 votes and the election would have been thrown to the House of Representatives (Democrat), so Gore would have won. The only way out was for the Supremes to interfere, which of course required a rather broad interpretation of the Constitution rather than a strict reading of the text.
** There's never just a 3rd party. There's always a 4th party that represents the disgruntled voters who aren't represented by the 1st, 2nd, and 3rd party. And the 5th party who are disgruntled by the 1st through 4th. Zeno's paradox of parties: there's an infinite number and you can't ever get out the door.
***BTW: please don't use Lincoln as an example of a third-party presidential win. The GOP started by taking over local and state governments when the Whigs imploded over slavery in the latter half of the 1850s. Before Lincoln was elected the GOP was already the second-largest party in the House of Representatives. That is, before Lincoln won they were already the 2nd party and the Whigs were demoted to an irrelevancy, so it was effectively a 2-party election in 1860.
**** Actually, until demography has time to play a role and turn Texas purple, my vote in the winner-take-all system is entirely wasted. It doesn't matter who I vote for because this state is going Republican, and so the electors representing me (and about 42% of the state) will be voting against our wishes. This blog really only applies if you live in a purple state where the vote is close enough that your electors could swing either way.