How do populism and populist leaders fit in with the rise of illiberal democracies?

Populism and populist leaders fit in with the rise of illiberal democracies in that they are the agents of their creation. Recall that an illiberal democracy can be representative, can have checks and balances, has elections, and can function under a robust rule of law. The difference between illiberal democracies and others is in how populist leaders and their enablers use those democratic institutions. Discriminatory or restrictive laws or policies about immigration, racial or gender equality, citizenship and access to resources are hallmarks of illiberal democracies. These laws or policies are made in the usual way, by deliberative legislation, judicial review and executive implementation, but they have effects which in other democracies are considered corrosive to society.
The role of populism and populist leaders in all this is in the creation, the culture, and providing of leadership. They focus the energy of a movement to take over a government, to restrict immigration, to harass marginalized groups, or even to perpetrate violence. Consider that a set of ideas without anyone pushing them usually don't have much effect. However, even a small idea with a big champion, or one that's attached to a very popular movement, can have very large effects.
Populist movements now holding or contesting power in the US and Europe fit this mold. Without figureheads or entrenched movements, populism dies. If Donald Trump weren't President of the United States, there would still be populist and illiberal policies, but they wouldn't have national legitimacy. They wouldn't be "official" attitudes of the country, and it would be harder to turn them into real laws or policies. The same is true of Hungary without Viktor Orban, Italy without The League, France without Marine LePen, and Germany without the AfD.
Populist movements and populist leaders are the vehicles which bring illiberal democracy to societies which think they could never have it, and they're the lifeline which sustains illiberal politics in government.