Before leaving Francine's apartment, you can steal health pills (1) and a sports watch (2) from the china cabinet behind her.
You'll have no way of pawning the watch now, but taking it unlocks the "Kleptomaniac" achievement/trophy.
Exit the apartment onto the balcony. Turn right and jump across the gap to grab the next ledge. Pull up. (You'll get the ammo down below shortly.)
Vault over the low railing. Climb the drainpipe...
...traverse to the right, and pull up onto the ledge.
Either run without hesitation or carefully jump over the crumbling sections of ledge ahead until you reach the next corner.
Turn around, hang from the edge, and traverse to the right until you can drop onto the small balcony below the red wire.
There's a drainpipe in the corner that leads down to the COURTYARD below, but you're missing a strength upgrade that will let you barge open the door there. So ignore it for now.
Stand beneath the red wire and jump up to grab it. Traverse across the courtyard and drop onto the next balcony. (If Lara won't immediately traverse forward, try pulling her legs up using the Duck button.)
Follow the balcony around to the end and safety drop onto the mausoleum roof below.
Hop down to the ground and follow the path toward the big wooden door.
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