On June 27, 2013, the senate passed a comprehensive immigration reform bill by a vote of 68 to 32 (14 Republican senators voting "Yea".) House Speaker Boehner refused to allow it to be voted on in the house.
There might be more to the Republican caterwauling about Obama's autocracy if they were actually democratic themselves. Indeed, were Boehner to allow a vote, an immigration reform bill would almost certainly be voted down by the Republican house majority. He doesn't want to because the debate and its press coverage would show that the Democratic proposal is the only rational, plausible and fair solution to the problem and that the Republicans are bankrupt of alternate ideas and are just grandstanding since doing nothing (the standard Republican "plan") will make life so much worse for everyone.
There is no real doubt that the president's act is an appropriate use of executive power. Republicans are just annoyed that they can't sabotage him on this initiative as they have on so many others.