What was Snape actually doing at 12 Grimmauld Place when he found Lily's letter and photo and when was that?

Severus went to Grimmauld Place immediately following his fulfillment of Dumbledore's last request. After killing Dumbledore, Grimmauld Place was the only place he felt safe, and he knew it would be the last chance he would ever have to search Sirius' room. Remember that Sirius had been James' best friend; Severus was hoping for some kind of memento, some little bit of Lily, because the house had never been cleaned out after Orion and Walbergia (Sirius' parents) had died. He didn't know it would be there, but he hoped.