To answer the specific question - on a national level, this would require a constitutional amendment, as the US constitution spells out the mechanism of elections.
Same goes for state level elections - would require a state constitutional amendment.
Both are fairly involved processes, requiring a lot of yes votes from a lot of voters.
And now, for practical reality:
In the US, the current political layout of two dominant parties would tend to make a runoff style election almost meaningless. The two current parties represent very different political outlooks, and any other parties draw almost no votes. So a runoff between the two would produce about the same results as the initial election. Do you see a voter switching from repub to dem or vice versa, just because the vote was close? Probably not. And the small third parties: libertarian, green, etc... don't draw enough votes to tip an election in anything but an extremely close margin.
In other democracies, such as the UK or Germany, two parties don't dominate. It is quite possible that during a runoff, a Green party supporter might swing to a Liberal Democrat, as they are similar in philosophy, especially if the Liberal Democrat made some commitments to the Green platform to draw their support.
Looking back at the 2016 US election with two very odious choices, a third candidate might have actually made some headway. Sadly, the only third party that had any traction at all, the Libertarian party, fielded a complete moron. (what is Aleppo?) Oops.
So, to answer the question in a more pragmatic manner: to get a runoff style election in the US that actually has meaning, the electorate would have to break the two party mindset. Enough voters would have to be willing to support other parties in sufficient numbers to add meaning to a runoff election. Given the two primary choices in the 2016 election, that's no longer a far fetched scenario.