We did it for $15000US (yes, that's right, 15 thousand) from the US, (MUCH higher airfares) and there were three of us, and I chose to be sure of our lodgings in advance, which I found, after we arrived, could have cost us about a third of what we paid, had we chosen to just hope for empty rooms, but with my husband and young son, it made sense to book in advance. $6000? You might be able to do it on a shoestring, and without reservations (they charge less if you walk in and they have an empty room). Don't forget the fuel costs...It cost about .95€/liter last year...I don't think it's as volatile as the US, though...that was diesel.
I would have a little extra set aside if possible, just to be sure you can make it.