Officially Published Routes for 5e
According to Waterdeep: Dungeon of the Mad Mage, an official published source, there are two ways to get to skullport (page 303):
Through the sea caves
This appears to be the best known way, with access to the sea that smugglers would need anyways.
Skullport can be reached by traveling through a series of natural caverns and magical locks that stretch from the South Sea Caves on Waterdeep's shore to the River Sargauth far below Mount Waterdeep. (...) The waters grow calmer as one travels deeper into the mountain, where a series of magic locks control
the water level. Vessels descend hundreds of feet as they pass between them. Where the locks end is a vast cavern containing a smooth stone retaining wall that rises 10 feet above the water's surface. The partially collapsed remnants of a ten-story-tall hoist loom above the retaining wall.
The Grand Hoist mentioned in the accepted answer is part of the Sea Caves route.
Skullport is connected to level 3 of Undermountain by the underground River Sargauth and by five meandering subterranean passageways: Beggar's Rest Pass, Shadow Pass, Steamfall Pass, Taglath's Gap, and Whisperhaunt Pass. See maps 25 and 26 in this section for the locations of these routes.
There also is a way to teleport out of skullport via teleportation circle, however this by default is one-way only. That is because this teleport is afforded as a service by a resident of Skullport for payment (page 308):
the owner of the Poisoned Quill, Tasselgryn Velldarn, who makes money from visitors needing to leave Skullport in a hurry. (...) If the characters come seeking an escape from Skullport, Tasselgryn offers to cast the teleportation circle spell but demands payment of 50 gp per party member up front. She knows the address of another teleportation circle in Waterdeep (in the attic of an orphanage in the Dock Ward) and sends the characters there if they don't have a teleportation circle address of their own.
Another entry (on page 309) states that she "has access to a teleportation circle", which could mean either a permanent circle, or mean that she has access to the spell to create such a circle on the fly.
It does not matter though, as Skullport is under the same “no teleporting in or out” ban that covers Undermountain, with this NPC having been granted exception by Halaster. It would not work for others to teleport in, even if they knew a circle in Skullport. (Thanks to @illustro for pointing this out.)
Spiral staicase from Castle Ward sewers
Waterdeep: Dragon Heist offers a further route, via the Xanathar's lair, which is connected to both the city of Waterdeep and Skullport (page 102):
Characters are most likely to enter Xanathar's lair by this route: a spiral staircase accessible from the Castle Ward's sewers and hidden behind a secret door. This staircase circumvents a more difficult route through Undermountain and Skullport.
This magically lit tunnel extends 300 feet eastward off the map. It ends at a staircase that climbs 20 feet to the Guts & Garters Inn, an establishment located in the sub-terranean town of Skullport, which is under Xanathar's control.
The corkscrew staircase mentioned in the accepted answer as connecting to the cellars of Castle Waterdeep is not mentioned. It might either be a separate access from earlier editions, or it might refer to this spiral staircase that starts in the Castle Ward sewers (not the castle cellars). According to the adventure, various factions such as the Harpers, Lord's Alliance, Emerald Enclave and Bregan D'Aerthe (and of course Xanathars minions) all know where to find this secret staircase.
Location of Skullport
Location on the map: The distance of Skullport to Waterdeep or to Undermountain is actually never explicitly mentioned, but if the spiral stairs down to Xanathar's lair start in the Castle Ward sewers and decend vertically, and the route from there to skullport is only 300 feet of tunnel, then it would likely be also under or very near the castle ward. As the Yawning Portal with its access to Undermountain is also located in the Castle ward, the distance between Undermountain and Skullport likewise can probably not be very large, given the size of the Castle Ward, under one mile.
Depth below ground: The first level of Undermountain lies about 140 foot below the basement of the Yawning Portal, the stairs to level 2 descend 200 feet (p. 26, DotMM), the stairs to level 3, the Sargauth level connecting to Skullport are not given a descent but if they are about 45 degrees steep, from their lenght on the maps they might go down another 200 or so feet, for a total of about 540 feet down. While there needs to be incline for the Sargauth to flow towards Skullport, the river is describes as "the sluggish River Sargauth" (page 48), it can't be too steep. Given that it also cannot go more than a mile, this would put the ground floor of Skullport somewhere between maybe 600 and 800 feet below ground, maybe 1,000 feet if the river meanders a lot, or there are falls and rapids.