Direct answer: the most direct route into the Deeproot Depths is a coffin you physically pick up in the Siofra Aqueduct, in eastern Siofra River, after defeating a pair of Valiant Gargoyles. You carry it, and you use it. There is also a second route through Leyndell that most guides never mention, and it is the one worth knowing.
Route one: the coffin from Siofra Aqueduct
Clear the Valiant Gargoyles guarding the Siofra Aqueduct, and take the coffin hidden there. Interact with it and you are transported into the Deeproot Depths, arriving in shallow water just across from the Great Waterfall Crest Site of Grace. Activate that grace immediately — the region is dark and the sightlines are poor.
The critical mechanical detail: the coffin is an item you pick up and carry. It is not a menu option. Players who walk past it, or who assume the area is linear and headed the other way, frequently never find it.
This is the route to use if you came down through the Deep Siofra Well in Caelid, since that entrance lands you near the aqueduct.
Route two: under Leyndell, and why the common claim is wrong
You will see this stated constantly: Deeproot Depths requires defeating Radahn. That is not true of the Leyndell route.
The alternate path runs through the Subterranean Shunning-Grounds beneath Leyndell. After defeating Mohg, the Omen in the Cathedral of the Forsaken, there is a hidden passage behind his reward chest. Follow it down to the Frenzied Flame Proscription grace, then roll into a southern wall to reveal another hidden path, and a second obvious hidden wall beyond that. You come out in the Deeproot Depths.
So Deeproot itself does not need Radahn. What Radahn gates is Nokron, which gates the Fingerslayer Blade, which gates the northern half of Ainsel. Those are three different locks and they get conflated constantly.
One more route-specific note: the Great Waterfall Crest area is reachable only from the Shunning-Grounds side. If you came in by coffin, that side is closed to you.
What to do on arrival
The confirmed Sites of Grace are Great Waterfall Crest, Root-Facing Cliffs, Across the Roots, Deeproot Depths, Prince of Death’s Throne, and The Nameless Eternal City.
Torrent works here, fast travel works, and flasks replenish on group kills. There is a Map Fragment on a corpse in a gazebo amid the destroyed ruins — worth taking, because the region sprawls.
This is the source of both the Ainsel and the Siofra rivers, and the furthest downward point of the Erdtree’s root network. If you have not taken the Deeproot Depths map fragment, do it here rather than relying on memory.
Evidence status
Research-reviewed. The coffin route from the Siofra Aqueduct, the Valiant Gargoyles prerequisite, the arrival point at Great Waterfall Crest, the two hidden walls on the Leyndell route, the Great Waterfall Crest exclusivity, and the Sites of Grace list were each cross-checked against multiple independent references. The claim that Deeproot requires Radahn was checked and rejected as inaccurate for the Leyndell route. Not gameplay-verified on camera.