Sir Robert is handsome and intelligent, he's a good orator and has a clean record. His wife is equally impressive and his marriage is sound. Even his colleagues across the aisle respect him.
But there's a very ugly skeleton in his closet. He made his fortune and accumulated power through dishonest means. He used his access to secret information to advise Baron Arnheim. Based on this insider information, the Baron then invested in the Suez Canal, swindling millions of pounds for himself and some for his informant, Sir Robert. That's called insider trading, and its illegal. (You can think of famous cases of real people who have had to do jail time for insider trading, like Martha Stewart, for example.)
Sir Robert's crime remained a secret for years, though, and no one had to know about it until Mrs. Cheveley entered the picture with her plan to turn a profit. Suddenly Sir Robert is faced with a dilemma and his moral flexibility shows itself. At first he refuses to change his position and support the Argentine Canal, despite Mrs. Cheveley's threats of blackmail. However, as Mrs. Cheveley paints a vivid picture of his ruin, Sir Robert begins to waver. He considers endorsing a project he knows to be corrupt, a project he would otherwise reject, in order to prevent Mrs. Cheveley from publicly revealing his secret.
We see Sir Robert's adaptable nature even before his will is tested. For one thing, he's a linguistic chameleon. Look at the language he uses with Mrs. Cheveley, versus that which he uses with his own wife. Like many politicians – indeed, many good communicators – he understands the art of speaking to people in their own language. In his public greeting of the flamboyant Mrs. Cheveley, he's all exaggerated flattery: "Everyone is dying to know the brilliant Mrs. Cheveley. Our attaches in Vienna write to us about nothing else." With his earnest wife, his style is more serious: "Gertrude, truth is a very complex thing" (1.359). He's even serious when he lies: "There is nothing in my past life that you may not know" (1.375).
As a politician, Sir Robert seems like a moral contortionist. He sees a situation, figures out the best possible outcome, and bends himself to make it happen. This moral flexibility comes as a surprise to his wife, who had previously put him on a pedestal and considered him to be absolutely perfect in every way.
Sir Robert's inner conflict is written all over his face. Oscar Wilde goes into great detail describing what he looks like (Check out "Character Clues: Appearance and Dress"). Sir Robert's face suggests "an almost complete separation of passion and intellect, as though thought and emotion were each isolate in its own sphere through some violence of will-power" (1.52). He's a man divided. On the one hand, he wants to fulfill his wife's demanding image of the ideal man, noble and honest. On the other, he's drawn to power and wealth. As Sir Robert admits to Lord Goring, he doesn't regret the actions that secured his wealth and position in society.
Sir Robert's inner conflict allows Wilde to explore the idea of an imperfect public figure. While Sir Robert's done some oily things, Wilde paints a vigorously sympathetic portrait.
The young Sir Robert was disadvantaged and ambitious. "I was twenty-two at the time, and I had the double misfortune of being well-born and poor, two unforgivable things nowadays" (2.21). The Baron Arnheim deal gave him access to what otherwise would have been off-limits – prestige, money, and access to the upper echelon of British Victorian society.
Now Sir Robert is trapped by expectation. The pressure to be perfect comes from his wife, sure, but also from everyone around him. An Ideal Husband is chock full of characters gushing over the seemingly impeccable Sir Robert Chiltern. Lord Caversham proclaims that Sir Robert has "what we want so much in political life nowadays – high character, high moral tone, high principles" (4.194). The irony is that, in Sir Robert's opinion, he would have been no political life if his principles had not at one point been slightly less than high.