I am not a Twitter kind of person and don't know very much about tweeting, but I don't think "retrato en clave" is an actual Twitter term (like hashtag); I can't find any sign that it is. I know, however, that along with verbal tweets people post photos of themselves. I think "retrato" is literally a picture of themselves that they posted, along with a tweet or tweets about their meeting. It doesn't sound very natural to call it a "portrait", but if you say "showing them", it makes it clear that it's a picture (or portrait) of them (as most pictures on social networks are).

"En clave (de)", as Jenni says, is a bit like "en plan de". It means in a certain style or tone: "en clave de humor" means in a humorous style or tone (this example is given in Collins).
http://www.wordreference.com/es/en/translation.asp?spen=clav...
The DRAE definition is:

"clave
en ~ de.
1. loc. prepos. Con el carácter o el tono de. En clave de humor."
http://lema.rae.es/drae/?val=clave

So "en clave “soy feliz”" would mean "in an 'I'm happy'" style". And I think it refers to the picture and to the impression it conveys, rather than they they posted the actual words "soy feliz". They tweeted an account of their meeting and some "piropos" (mutual compliments), together with a happy smiling photos of themselves to show what a great time they had. "Soy feliz" is the message the picture puts across, the mood or tone (clave) of the picture.

I can't see a convincing way of doing "en clave “soy feliz”" at all literally, so I'd suggest something like "all happy and smiling":

"Louise Renault, all happy and smiling in Russell Square."
http://www.flickr.com/photos/kevinsteinhardt/2473831067/