If you want to get rid of the Democrats, your best bet lies not with the Populists, but with their urban brothers, the Socialists. But certainly a stronger Populist movement would help.
Let's say that William Jennings Bryan is passed over for the Democratic nomination in 1896 and the Convention instead decides to renominate Grover Cleveland. Cleveland loses to McKinley, and the Populists run Weaver again. Weaver gets a respectable share of the popular vote and wins much of the west in the electoral college. Four years later, McKinley wins re-election and the general prosperity calms down the Populist vote, though Democrat George Dewey still does relatively OK owing to his position as an admiral in the SpanAm war.
McKinley doesn't bite the bullet (pun intended) the way he did IOTL. The Socialist Party is growing by this point and the Populists, splitting into various factions over the way to go forward (left Populists favor and alliance with the Socialists, right Populists favor continuing on as an independent force) agree to a relatively painful divorce that leaves the right of the party nominating William J. Bryan for the White House in 1904 and the left supporting Eugene Debs. Democrat Alton Parker gets beaten by Charles Fairbanks, though by this point it seems that the rapid growth of the Socialists will allow them to outpace the Democrats at some point in the future.
Keep up the growth of the Socialists with some more conservative Republican leadership, and the Democrats turn out a lot like Britain's Liberals did, as the third (minuscule) party lodged between conservative Republicans and the Socialists.