How does Iago cast aside suspicion of his own part in the plot?

For the most part of the play, he doesn't actually have to, because - and this is crucial to reading the play - his manipulations are all done through playing mind-games with the other characters. In terms of his plot, there are few actions he actually performs.

First, of course, he plants a seed of suspicion in Othello's head that that Desdemona and Cassio are having an affair. Iago then organises Roderigo (who he has deluded precisely as he has deluded Othello - Roderigo thinks Cassio and Desdemona are sleeping together) to attack Cassio. Cassio fights back and is stripped of his lieutenantship by Othello. Iago, of course, remains "honest", and the friend of both men. And Roderigo has done the dangerous bit.

Desdemona now, without any steering from Iago, pleads to her husband for Cassio's reinstatement, which only increases his suspicion. Next, Iago has Emilia steal the hankerchief (she tells us he's begged her to steal it many times) - and he plants it in Cassio's room. Again, someone else is dispatched to actually do the dirty work. And finally, when Iago has Othello overhear Cassio and Bianca's conversation, that hankerchief provides the crucial "proof".

So, only Emilia and Roderigo have any reason to suspect Iago! His solution to that problem is to kill them both: though he kills Emilia too late! No-one sees Iago doing anything because he hasn't done anything.