Père Lachaise is indeed a wonderful cemetery in Paris. But my favorite cemetery in Paris doesn't have any dead people -- it's got dead animals.
It's the Pet Cemetery ("Cimitière des Chiens") in Asnières, just across the river from the Paris city limits and easily reachable by Métro (the Gabriel Péri/Asnières stop on Line 13). It's a huge and beautiful place, with lots of people coming by daily to place fresh flowers at this grave or that, plenty of interesting tombstones (e.g., "Deceived often by men, but never by my Frou-frou"), and even a celebrity (Rin Tin Tin).
And, yes, a perfect place for a picnic.