If you helped the Beggars they will have riot going by now. This will clear out the vast majority of the guards here and make your life much easier.
Head down the hallway you start in then go over to the left. Head through the tunnel past the split to the first junction. Turn to the right there and head down this tunnel. It will lead to a chamber with an Extremist inside it. Wait for him to pass you by then take cover against the nearby stack of barrels. Use this to get behind him and take him out. Just keep going down the tunnel after it. Go up to the lip of the drop ahead of you.
In the area blow you there are 2 Extremists to deal with. Go up to the lip carefully when neither are looking. Jump out onto the pillar in the middle of the room. You can easily air assassinate one of the guards then use a Quick Shot with the Phantom blade to take him out or just move in for a Running Assassination. That or wait for them to both turn and then jump down and head through the next passage on the left.
This leads to a chamber with more Extremists inside it. You can either fight them or climb up the pole in the middle while they are looking away. At the top go over to the right and toward the center of the catacombs. This will trigger a cutscene with La Touche and your target.