The key is to improve the Liberal Republicans' performance in the 1872 presidential election -- and that means a better candidate than Horace Greeley. Someone like Charles Francis Adams, perhaps?
They don't need to actually win in 1872 -- they just need to perform well enough to be taken seriously as a new opposition party, so they can supplant the Democrats (who were seriously weak at that point). In OTL, the Democrats made a massive comeback in 1874 in the midterm elections due to dissatisfaction with the Grant Administration -- in TTL, that support goes to the Liberal Republicans instead and leaves them in control of Congress.
At this point you'll find many Democratic politicians (especially in the North, of the Bourbon Democrat mould) defecting to the Liberal Republicans when they see which way the wind is blowing; the Democrats may hang on a bit longer in the South as a regional party for state politics and some Congressional races. Among these defectors could be a certain Northern Democrat named Samuel J Tilden; if the Liberals have him as their candidate in the 1876 election, they could win the Presidency.