Why is Claire at Clean Water Initiative?

Initiatives like that, have a huge impact on politics. With his wife running the CWI, it makes Frank look good before the general public. If it were ever to come down to the people voting for him, people would favor somebody who supports such initiatives. It's like brownie points in politics!

I didn't see anything in the episodes (at least which jumped out) spelling this out. The way I see this helping the symbiosis is she needs to be doing something which will not interfere with anything he is doing. Being the Director of the non-profit shows her willingness to support bigger things without any interference. It's kind of like the First Lady, Michelle Obama doing stuff with children's health: it helps their husbands look better. They don't have to do these things, but do so to further the overall goals of the administration. It's all politics.

I think that are 2 parts to answer this question:

• The first is simple and already pretty well explained here: To make Frank look good. Like a First Lady, Claire workes on CWI to improve Franks public image by showing that his family care about social-politic topics that are not directly related to the topics that Frank deals daily.
• The second one is a little more subtle. Nothing in the plot works as a "hard evidence", but there are subtle facts that can suggest that. And knowing how things works in real life makes me think that this could be the way CWI would be useful for the Underwroods. I think that CWI can work as a dummy corporation for the underwoods as income money from the gas lobby. Remy Danton offered to invest in CWI Africans endeavors, and at the same time Frank was worried that this could implicate in him owing some favors for SanCorp. This can be the start-point of a corruption scheme where the Underwoods laundry money trough CWI in return of political favors for SanCorp. This corruption scheme doesn't seem to be in place at the time of the series but you can infer that it can have happened before since Frank, Claire and Remy seem to have a relationship for a while using CWI as a means to an end.

[Claire and Frank arriving at the symphony]
Frank: I'm curious. Are we patrons, benefactors, or lepers?
Claire: Believe it or not, we're angels.

It's all about image and power. Yes, she founded a non-profit charity to make had husband look good, but if you recall, their marriage is very much a partnership. The higher Frank goes, so does Claire. With Frank in power, it gives Claire a springboard to rise. And coming into power together was always their intentions. However, the Underwood name would only do so much. She may have notoriety because of Frank, but in order to hold any position of power in Washington, she would have to have some kind of legitimacy to win over the American people. Creating a nonprofit foundation is one thing. But like she said in season one they've plateaued. Anyone can create a business, but expanding and evolving is what makes people truly successful. Claire needed the money to do that for CWI. To add legitimacy to her success. Whether she cared about clean water or not, who knows. She definitely wasn't passionate about it like Gillian Cole from World Well. She was more interested in seeing the organization move forward and grow.