During the campaign for Frank's presidency in House of Cards, it always looked like people were more connected to Claire than Frank. We also heard Frank saying "You are right, this office has only one chair and you have always known that." Does Claire Underwood want the Presidency for herself?
Yes. In fact, Claire doesn’t just want to be president, she has a deal with Frank that she will run for office at some point with his support.
Frank and Clair’s deal is made clear in a scene in the first episode of season 3, where they have a middle of the night conversation in the White House. Claire doesn’t want to wait with the announcement of her nomination as ambassador, because “the timing will never good” while she doesn’t “want to waste time” when she can be “getting real experience under [her] belt.” When Frank tries to respond she interrupts him to say they “made a deal” that he’s now “trying to back away from.” Frank argues that he can’t make the announcement yet, to which Claire replies that they “agreed to this before we ever stepped foot in this house.” Just what they agreed to becomes clear when she then asks: “How am I supposed to run for office at some point if I don’t have a legitimate track record?”
The scene plays out near the end of the episode, at around 48 minutes in:
Claire: Francis. I want you to announce the nomination this week. I don't want to wait. The timing will never be good. And I don't want to waste time when I can be getting real experience under my belt. So set a date, hold a press conference, and let's do it.
Frank: Claire! I was not trying to suggest that you not be the ambassador - when I brought up Cathy. I was just--
Claire: We made a deal, and now you're trying to back away from it.
Frank: That is not true.
Claire: Then you would've told Cathy it's happening to make her peace with that.
Frank: It's not that simple.
Claire: Yes, it is. You're the president.
Frank: I can't control Congress. I'm trying to get two major initiatives off the ground and every move I make will have an effect.
Claire: We agreed to this before we ever stepped foot in this house.
Frank: I know what we agreed to.
Claire: How am I supposed to run for office at some point if I don't have a legitimate track record?
Frank: We have to survive the next 18 months, accomplish something, and then look ahead.
Claire: Francis, let's be realistic. There's no guarantee you'll get elected. I need to be prepared if you don't.
Frank: You have that much faith in me, do you?
Claire: More faith than you have in me, it seems.
Frank: I believe in you more than anyone, Claire.
Claire: Then show it. Nominate me. I shouldn't have to convince you.
Frank: It's not about convincing. It's about being honest about the political climate - that we find ourselves--
Claire: I'm almost 50 years old. I've been in the passenger seat for decades. It's time for me to get behind the wheel. That needs to start now. Before the election. Because who knows what will happen? What if you lose?
Frank: I will not be a placeholder president, Claire. I will win. And I will leave a legacy.
Claire: You mean we will.
Of course, the deal is that Claire will run for office after Frank, but given the ending of season 3, there’s some speculation that the plot for season 4 might just involve Claire running against Frank in the ongoing election.