Apparently she cheated on her first husband on the set of Daredevil with Ben Affleck (filming was in 2002, they filed for divorce in early 2003). But Ben didn't want a relationship. But her ex-husband later said this wasn't true, that they split up because she became a celebrity/famous and he didn't like the spotlight.
Anyway, she started dating Michael Vartan, her co-star on Alias a few months after filing for divorce (summer 2003). It was relatively high profile because their characters also dated on the show. Then she cheated on Michael with Ben in 2004, which was just before Season 4 started filming. During Season 4, Michael and Jen's characters were engaged on the show, so that made everything awkward. And supposedly Jen asked the producers to limit Michael's screen time after that so in Season 5, his character appeared in only 5 episodes (and they only brought him back for the last 3 because of fan reaction. The fans insisted his character be brought back for the series finale). This was also the season Jen got pregnant with Violet, which they wrote into the storyline for her.
I can't really verify the first story because I didn't follow Felicity; this was just rumored during her time on Alias. But the Alias stuff was talked about on Alias messageboards in 2005. Her reputation as a social climber is probably because within months of a split, she starts dating someone more famous.
And her publicist is amazing because everyone still sees her as sweet Jennifer Garner.