The electoral system is defined in the constitution (Article II and Amendment XII), so to completely abolish it would require a constitutional amendment. That can technically be done in a couple different ways (see Article V), but in practice requires two thirds of congress and three fourths of the states, so it's very difficult. Still, such amendments have been proposed, the most recent being filed today.
Your second suggestion (or something like it) is much more feasible. The National Popular Vote Interstate Compact (NPVIC) is a state law that awards all of a state's electors to the winner of the national popular vote, but only once enough states have joined the compact to make an electoral majority.
Article II stipulates that legislature of each state direct the selection of the state's electors, so they're well within their power to change that mechanism, and the NPVIC only requires enough states to represent a majority of the electoral college, which is a much lower bar than the three fourths for an amendment.
At the moment, it has 165 of the 270 electoral votes needed to take effect. However, crossing the 270 threshold will require some states that are currently Republican-controlled. As only Democrats have ever been burned by the Electoral College, Republicans don't have a lot of incentive to change the status quo. However, most Americans do support a national popular vote for president, so perhaps the political will is there.