- Are you now both a lich and a vampire? No, you don't exist because there aren't illithids in Pathfinder. Also, no, because undead creatures are immune to mind-affecting effects, making the power true mind switch unable to target either the illithid lich or the illithid vampire.
- Do you have flesh? Yes. An illithid has slimy flesh. A lich has withered flesh. A vampire has cold flesh. So if--somehow--this combination succeeds, the end result would likely be the cold, slimy flesh of the vampire illithid instead of the previously slimy, withered flesh of the lich illithid.
- How do you reform if you die? The text of the power true mind switch explains that the last body a creature inhabits is its "true" body. The phylactery would likely recreate that one.
More details below.
The Scheme Has Issues
There are several. I'll address them in the order they appear in the question.
A proper, self-respecting illithid...
...doesn't exist in Pathfinder. Illithids were one of a handful of creatures--including beholders and yuan-ti--that Wizards of the Coast decreed Dungeons and Dragons product identity (somewhat after-the-fact in some cases, leaving a few publishers who had published books about such creatures forced to destroy remaining copies, if I remember correctly). They aren't part of the SRD, and no one can include them in their games without permission without risking reprisal, the same way you can't include in your game vulcans or hobbits. You can read more about this here and here.
...is a psion[, most likely a] telepath.
Okay. Sure. That's a reasonable choice for an illithid.
At some point, [it] should get [the power] true mind switch.
So if 17 levels of Pathfinder telepath gets stapled onto the typical mind flayer from the Monster Manual for Dungeons and Dragons 3.5--which is what must be used because there is no Pathfinder mind flayer--that mind flayer should be expected to take this specific power. Also, so we're clear, that's a CR 25 creature, appropriate as a boss battle for a party of level 20 Pathfinder PCs. (Epic level play isn't supported directly by Paizo.)
But fair enough.
[Then the illithid got] into necromancy and became a lich. [Its] soul is now in [its] phylactery.
Assuming psionics-magic transparency, an illithid telepath 17 could certainly create a phylactery if it meets the requirements, granting it the template lich. This however, changes the illithid's type from aberration to undead. This change is really important.
[The illithid then gets] a vampire to create a spawn out of [another] illithid then [the first illithid kills] the vampire.
I can concoct a scenario in which cunning, vicious illithid might convince bumbling vampire to turn lackwit illithid into a vampire, and afterward kill bumbling vampire. The underdark's a wretched hive of scum and villainy. But mechanical problems remain.
An illithid in Dungeons and Dragons 3.5 is an aberration, and that game's vampire template excludes aberrations from gaining the template vampire, restricting the template to humanoids and monstrous humanoids as per the SRD. (Other texts modify this, but let's keep it simple.) Pathfinder's template vampire isn't restricted in the same way, allowing aberrations to become vampires. So if mind flayers in a Pathfinder game were not sufficient evidence of house rules, illithid vampires are certainly sufficient evidence the DM is mixing systems and making up stuff. (You should ask what Dungeons and Dragons 3.5 things are available to PCs.)
(Also, a vampire spawn is a specific kind of creature created by vampires from low Hit Dice creatures; those Hit Dice are exceeded by the mind flayer. Hence I'm pretty sure a mind flayer who becomes a vampire would become a free-willed vampire. And, in all likelihood--unless the plan's been cleared with him first--an angry free-willed vampire.)
[The illithid lich uses the power true] mind switch [and transfers his consciousness and soul] into the vampire illithid.
The big problem here is the power true mind switch is mind-affecting and both the lich illithid and the vampire illithid are immune to mind-affecting effects because they're undead.
There's at least one way around this, and I'm sure there are others, too, but this creature is CR 25 before trying to become a lich vampire illithid. Giving him more class levels just makes him more unlikely to ever see play.
Finally, Pathfinder is very clear about what abilities one gains when one switches bodies using the power true mind switch (unlike the Dungeons and Dragons 3.5 version of the same power, which reads like it was never designed to see play) and excludes the fun stuff like spell-like and supernatural abilities. I guess the lich illithid that manages to pull this off might end up with slightly better statistics and the ability to spider climb, but that's a really convoluted set up for relatively minor gain.