I've found that doing rifts with a good party is the fastest way. I currently have more than 8000 unused Forgotten Souls. I usually clear T10 rifts and salvage every legendaries that is not usefull for my character.
If you really need Forgotten Souls and have plenty of other crafting materials, you can use Kanai's Cube to upgrade rares into a legendaries and salvage them. Moreover, the bloodshards you get from the Rift Guardian allow you to gamble at Kadala, significantly increasing your legendary 'drop' rate. Farming in party will help you farm faster and in higher Torment resulting in more Forgotten souls.
Pick the higher Torment level in which you can complete the rift quickly (2-3 min max). Doing rifts : The rift guardian has a high chance of droping a Forgotten Soul.There are several ways for getting Forgotten Souls :
