Clearly, Lyndon Johnson did deliver, especially on the issue of civil rights. One can argue that Johnson got more done in this area than Kennedy ever would have been able to had he lived.
Johnson helped to bring into being the two most important civil rights laws of all time. These were the Civil Rights Act of 1964 and the Voting Rights Act of 1965. These laws effectively gave African Americans the rights that they were always (since the Civil War) supposed to have.
As far as poverty goes, Johnson was successful in getting a lot of legislation passed. His War on Poverty set up many programs that were meant to help the poor. Of course, he did not actually eradicate poverty. However, he did deliver the legislation that, he thought, would eliminate poverty. In this way, he delivered in that area as well.