How much is Claire Underwood aware of her husband's more serious deeds?

First episode of season two shows Claire looking at the news of Barnes' death. Her momentary pause and small hand gesture indicate that she has her own thoughts about the death, most likely concerning Frank's involvement. Does she discuss them with Frank? Unlikely. It reflects their mysterious relationship. She knows, he knows, but it isn't said explicitly. In fact, we rarely see them having any substantial discussions. Their most interesting scenes end up being powerful tableaus. I see them as superficial partners connected by a common goal, but an intimate and emotional connection isn't evident. Why superficial? Because their facade still continues even in their own homes. I doubt that Frank knows anything about Claire's breakdown.

I'm limiting myself to what you've already seen up to this point. There are spoilery answer to your question, but I am avoiding those.

What we already know about Frank and Claire:

• Their marriage is not based on love, but actual (professional) partnership. It's more a two-man-company than it is a marriage.
• Frank did not feel threatened by Adam Galloway. Because Frank knows that what Claire seeks with Adam, Frank is not able to provide. There's no conflict, Claire just needs something more than what she has with Frank. As long as Adam is second to Frank, Frank does not care.
• When Frank was having sex with Zoë, it was implied that Claire did not know about it. However, when Claire is finally shown to be aware of the affair, she does not care one bit. It's never explicitly stated whether Frank kept Claire in the loop or whether Claire simply pretended to always have been in the loop in front of Zoe, but one thing is very clear: Claire does not feel threatened in any way by Zoe. She doesn't even seem to think that Frank's thing with Zoe is meaningful. Which, to Frank, it isn't. It's simply part of the plan.

Frank and Claire are working together. They have a plan. They are not a happily married couple (in the traditional sense), nor are they (predominantly) individuals with their own agendas.

So it stands to reason that Claire follows Frank's lead. She understands the political benefit of killing Zoe, and since the political benefit is their only goal, why should she be upset?

Imagine that you and me are bank robbers. Suddenly, I find out that you've killed a cop. Now, there are two options:

• I get upset at you, because killing a cop brings more heat on us. Why would you do that? (this is the reaction you are expecting from Claire)
• I understand that you killed the cop because he was close to finding out who we were. You killed him to keep us safe (this is Claire's actual reaction)

Claire is pragmatic, just like Frank. Therefore, they will pick the latter option. If they were the kind of people who picked the former option, then they would not have been as successful as they have already proven to be.

Claire's involvement with Frank's dealings will be elaborated on more in future seasons. You'll see.

To shamelessly steal from Reddit:

Someone noticed that Frank is looking at a map of the DC Metro System, right before pushing Zoe in front of it, when Claire walks in and says "I know you'll do whatever you think is best."

Given the context she is almost certainly referring to Zoe's death, and given her lack of shock at the idea of murder it seems likely that she has been aware of Frank's past murder(s?).

The fact that Claire and Frank sometimes boast about being completely honest to each other (affairs and all) pushes me into thinking she is fully aware of Frank's deeds.