Go down the stairs underneath the moondial. Your task now is to explore Castle Volkihar's Ruined Tower. Serana mentions she has never been in these tunnels before, but she bets they run right under the courtyard and into the tower ruins.

Climb up the stairs, and activate the pull chain to open a secret door. Climb some more stairs to the east and open another door. Kill the skeletons seated around the dining table. Climb the stairs to the south. Fight the Gargoyle Sentinel. Go through another door, then turn east and up some more stairs. There is a door here with an adept lock, hiding a small room behind with some treasure in it.

Climb further up on the stairs to the southeast. Kill another Gargoyle Sentinel. Keep going up the tower, then down into another big room. Arrive at a room with a gargoyle statue and a pull chain. Activate the pull chain and defeat the sentinel. Go up more stairs to the northwest and use another pull chain to open the gate. Keep working your way up, cross a bridge, and continue upwards. Go through another door and into a room with a table and some gargoyles.

There is a Vampire Royal Armor on one of the shelves in this room. This light armor has a good armor rating and makes your magicka regenerate 125% faster. Plus it makes your character look cool, so try it on.

Serana says that she does not think we have reached the top yet. She thinks there is some kind of secret passage around here.

Look for a candlestick on the southeast wall. Turn the candle to reveal the secret passage. (See screenshots below).

Skyrim Dawnguard Candlestick
Skyrim Dawnguard Secret Passage

Climb up and reach some sort of necromancy lab with a big circle in the center. Serana will tell you to look for clues. You need to locate Valerica's Journal. The journal is located on some shelves on the southwest wall of the laboratory. See pictures below for exact location of the journal:
Dawnguard Valerica Journal

Dawnguard Valerica Journal

Read Valerica's Journal. It talks about a portal to some area called Soul Cairn. Speak to Serana after reading the journal. Tell her you found her mother's notes. Ask what is that Soul Cairn that she mentions. Serana tells you that the souls inside soul gems don't vanish when they are used, but instead end up in the Soul Cairn. The Soul Cairn is home to very powerful beings. Necromancers send them souls, and receive powers of their own in return. Serana suspects that the circle in the center of the room is some type of portal. A formula in her mother's notes should give us safe passage into the Soul Cairn.

The formula requires a handful of soul gem shards, some finely ground bone meal, a good bit of purified void salts, and a sample of Valerica's blood. You tell Serana that she shares her mother's blood so that can be used instead.

Your objectives now are as follows:
Gather Soul Gem Shards
Gather Finely Ground Bone Meal
Gather Purified Void Salts

Soul Germ Shards
Above a cabinet on the east side of the room

Finely Ground Bone Meal
This is found on a table near some skulls

Purified Void Salts
On some shelves in the top part of the laboratory

Once you have all the ingredients, place all the ingredients in the vessel. Locate the portal vessel and fill it. Speak to Serana again. She adds her blood to the vessel. The Soul Cairn portal opens. Enter the Soul Cairn.

This completes the quest, Chasing Echoes, and starts the next quest, Beyond Death.