I happen to have a copy of Tales from the Yawning Portal, published after you posted this question, and it indeed contains the adventure module Dead in Thay. It is not quite the same adventure module as was used in D&D Next, with a few parts of the adventure omitted or modified. Regardless, the introduction to the adventure in Tales from the Yawning Portal, plus a few other details scattered throughout the module, contains a window into the activities and personalities of the Red Wizards of Thay.
I will admit up front that the information from this module is quite limited in scope, so probably doesn't constitute the 'best' source, but it is an official source current to 5E, and the only one I have. I will summarise the key points below. Some elements could be considered spoilers for the adventure, so I have put them in hiders. You will want to consider whether your adventures take place before or after the events of Dead in Thay.
We are given some history. Kazit Gul was a Thayan archmage who eventually became a lich. He engineered the Doomvault, a massive dungeon deep beneath the Thaymount, with the intent of luring in adventurers and harvesting their souls. As Thay became more hostile to outsiders, fewer people sought the Doomvault, and Gul became a demilich. At some point, Szass Tam and his followers defeated and enslaved Gul. The Doomvault has since been re-purposed it into an evil lair, where the Red Wizards perform experiments and build their armies.
Hidden beneath the Doomvault is the Phylactery Vault. Within the Phylactery Vault are the phylacteries of Kazit Gul, Szass Tam and all his elite lich servants.
By the end of Dead in Thay, the Phylactery Vault and all the phylacteries within are destroyed. It is ambiguous as to whether Szass Tam's phylactery was also destroyed, or whether it was somewhere else, as Szass Tam's phylactery is mentioned only once while all other mentions of the Phylactery Vault only refer to Szass Tam's elite liches.
At the time of the adventure, the master of the Red Wizards of Thay was the lich lord Szass Tam. We do not meet Szass Tam in Dead in Thay, but we are told a fair bit about his schemes. He is attempting to conquer the North then all of Faerun. He is also seeking god-hood.
We learn of another character: the lich Tarul Var. He appears to be a general, with Szass Tam as his master. Var was in command of Bloodgate Keep and the attempted invasion of the Sword Coast. After failing at Bloodgate Keep, Var fears Szass Tam, who will probably punish him. We are told a bit about Var's personality.
Devious and arrogant, Tarul Var thinks all other creatures are beneath him. He berates the characters at every opportunity, and he drives his underlings to cruelty.
As for Var's fate,
Tarul Var may or may not be dead at the end of Dead in Thay. This depends on whether the player characters kill him or let him live, and whether they destroy the Phylactery Vault before he rejuvenates.
The adventure module names a few Red Wizards who aren't liches, but they probably aren't that important outside the Doomvault.
There is one other major character among the Red Wizards, who reveals that Szass Tam's leadership is not absolute, and that is Syranna, a leader in the Thayan Resurrection resistance movement. From the information provided, the Thayan Resurrection appears to be run by Red Wizards who are dissatisfied with Szass Tam's leadership and are rebelling against him. These Red Wizards are probably regarded as renegades and rebels by other Red Wizards.
Syranna had some noteworthy words to say about Thay and Szass Tam.
"Many Red Wizards chafe under the rule of Szass Tam. Once, Thay was a land of learning and power. Now, death scours Thay while the lich lord ignores all concerns other than his quest to become a god. If his mad plans are left to run their course, none will be left alive here to worship him."
After the events of Dead in Thay,
with the Phylactery Vault destroyed and the Doomvault in disarray, much of Szass Tam's power is broken. 'She invites any characters who proved useful, particularly those with arcane talents, to remain in Thay under her leadership. She promises them power and influence if they throw in support for her.' From this, we can infer that Syranna plots to overthrow Szass Tam, and if successful she would be the leader of Thay after Dead in Thay, not Szass Tam. As lawful neutral, she probably won't be trying to conquer the world like Szass Tam. Probably. But she's still a Red Wizard, so they'll still be doing some Red Wizard things.
In summary, Dead in Thay tells us the following key pieces of information about the Red Wizards of Thay: Their leader is Szass Tam, a lich lord with plans for conquest and godhood. Tarul Var is one of Tam's generals, and also a lich. Working against Szass Tam is the Thayan Resurrection, a Red Wizard-led rebellion, with Syranna as a key leader.
