From the article:

In Iraq, Iran's chief ally has been the Badr Organization, formerly the paramilitary wing of what's now the Islamic Supreme Council of Iraq, the country's largest Shiite political party. During the Iran-Iraq war, Badr operated as a wing of the Iranian military; after the toppling of Saddam, Badr members infiltrated the security forces and were believed to be responsible for torturing and killing jailed Sunnis.

Did you know that the leader of the Badr org visited the White House and is one of our chief allies in Iraq?
http://www.whitehouse.gov/news/releases/2006/12/2…

As for this:

One of Suleimani's first major victories against the United States in Iraq, however, was the product of political shrewdness, not military force. It came in January 2005, when Iraqis voted for the first time since Saddam's ouster nearly two years earlier.

The Bush administration pulled out all the stops to keep secular, pro-Western interim Prime Minister Ayad Allawi in office, aiding him with broadcast airtime, slick campaign ads and veteran advisers.

Suleimani countered with a covert PR campaign on behalf of a bloc of conservative pro-Iran Shiites that he helped assemble, and he sent printing presses, consultants and broadcasting equipment, said a senior Iraqi official who's known Suleimani for years. He spoke on condition of anonymity because of the sensitive relationship between Iraq and Iran.

When the ballots were counted, Bush pointed to the purple-dyed fingers of Iraqi voters as a triumph for democracy — but Allawi and his bloc were out and Iran's allies were in.

Allawi had no native support, and you couldn't expect a secular person to win at that time, either. Plus he was just a big fraudster anyhow. And of course Iraqi Shiite groups that had been sheltered by Iran during Saddam's time won the elections, simply because the population is mostly Shiite.

J: "to subtly overthrow Iraqi democracy and liberty"

Maybe just the American overlay. Of course they want to affect things in Iraq. Can you imagine if Russia had invaded Mexico– you can be damn sure we would be making our desires known there and be deeply infiltrated (as well as simply having a lot of friends).