What meaning is given to Claudius' aside as they plan Ophelia's action in Act III?
Claudius essentially reveals his guilt for the first time here. Although the Ghost has told Hamlet that Claudius killed him, & Hamlet suspected something all along, this is the first proof we have from Claudius himself. This aside comes when Polonius and Claudius are plotting to catch Hamlet in his madness, & have used Ophelia as bait. They end their conversation, & Polonius instructs his daughter:
Ophelia, walk you here. Gracious, so please you,
We will bestow ourselves. Read on this book,
That show of such an exercise may colour
Your loneliness. We are oft to blame in this—
'tis too much proved—that with devotion's visage
And pious action we do sugar o'er
The Devil himself.
Thus Polonius, in one of his rare moments of clarity, remarks that they are perhaps doing something wrong. Essentially, they are trying to cover up sins through piety and nice actions, but they cannot truly hide their mistakes. Claudius then applies this to his own hideous acts:
O, 'tis too true!
How smart a lash that speech doth give my conscience!
The harlot's cheek, beautied with plastering art,
Is not more ugly to the thing that helps it
Than is my deed to my most painted word.
O heavy burden!
Polonius' words are like a whip (lash) to Claudius' mind. He then connects his attempt to hide his brother's murder to a prostitute putting on make-up. He argues that his "painted" words are uglier than any prostitute could ever be. this is significant because we know know (although Hamlet still does not) that Claudius is 100% guilty in his brother's murder. The next scene lends dramtic irony, as we experience Hamlet's moment of discovering, having already seen it for ourselves.