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. Walk carefully to the edge of the balcony, turn around, hang from the edge, then traverse right to grab onto the drainpipe. Climb down to the ground.
Barge down the door opposite the little doghouse. (In the original game, you needed a strength upgrade to do this, but in the remaster Lara already feels strong enough!)
Picking up the V-packer standard cartridges (3) in this alley unlocks the "Death-Defying" achievement/trophy.
NOTE: If you entered the graveyard via the Doorman's gate, you'll get the ammo and achievement/trophy a little later.
Climb back up the drainpipe to the little balcony below the red wire.
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. (This is where you would have entered if you came past the Doorman.)
Pick up a small medipack (4) behind a gravestone near the wooden door (i.e., on the right when facing it).
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