Due to the late arrival of Destiny 2 on PC, most of the game's little secrets and boss strategies are already out in the open. Because of that, there's a good chance a number of groups looking to run the Leviathan raid at launch won't be doing 'blind' runs console players had no choice in heading up. If you want to go in fully prepared, this speedrun cheat sheet should help.
Reddit user ItsGShark recently posted a speedrun strategy that netted team 'Euros' a completion time of 22 minutes and 10 seconds. The Leviathan raid itself is part of Destiny 2's weekly lockout timers, so those looking to blast out a bit of high-level content for the week can get it done in little time.
Remember; Destiny 2 fireteams may usually consist of 3 people for general PvE content, but the Leviathan raid accepts groups of 6. The player in question here took 3 Warlocks, 2 Titans, and 1 Hunter - with the Hunter being the first to rack up kills as they spawn into the encounter.
Then it becomes something of an exploit, having certain players rush forward and die through shortcuts so the rest of the team is teleported forward by being too far away from the leading player. The 'Royal Pools' fight barely changes here, and with The Gauntlet being mostly scripted, time saves are gained by simply no screwing up and being a little faster on the trigger. In the Pleasure Gardens, they opt to limit themselves to only 2 flowers before rattling off the dogs and heading to the final boss.
Being a light speedrun guide, this isn't the best for newcomers to read up on before heading in. Backed up with a properly detailed encounter guide, however, and you'll have a pretty good idea of the Leviathan raid's min/max situation. You'll know what's necessary, what isn't, and how to optimise your time in there. Above all else, it'll give you and your team a better sense of the strengths and weaknesses of your chosen class.
It's a shame we can expect a decent amount of Destiny 2 raid teams to ask that players familiarise themselves with a guide or raid tutorial video, but judging from other MMOs, if there's a guide available, players expect others to do their research to minimise time lost. People just want the gear over the experience more often than not.