Much of the backstory to Dooku's decision to leave the Jedi and join the Sith is explained in the Legends novel Darth Plagueis. Darth Plagueis and Darth Sidious both held numerous conversations with Dooku and Sifo-Dyas to influence them.
Dooku became convinced that the Republic needed to be destroyed and rebuilt in order to fix its problems. Dooku felt this was needed because the Senate was ineffectual and the Jedi Council was unwilling to unleash the full power of the Force (i.e. use the dark side) to fix the Republic's problems (unlike the Sith). Dooku was convinced that he could rebuild the Republic by working with Palpatine (he did not yet know Palpatine was a Sith Lord and had developed a political friendship with him) and the Sith. Moreover, he felt that the Sith were not pure evil but merely approached the Force in a different and less restricted way.
The relevant quotes from Darth Plagueis are as follows (I've bolded the particularly important parts):
Dooku adopted a sly smile. “I understand politics, my friend. I know
that you have to be circumspect about what you say and to whom. But
that the disenfranchised worlds of the Outer Rim enjoy any support at
all is largely due to you. You speak honestly and you champion the
underprivileged, and you may be the only one capable of bringing the
Republic back from the brink. Unless, of course, you have been lying
to me all these years.”
Palpatine made light of the remark. “Perhaps a few lies of omission.”
“Those I am willing to forgive,” Dooku said, “whether or not we become partners in addition to being allies.”
Palpatine interlocked his hands. “It is an interesting notion. We
would have to deepen our conversations, become completely honest with
each other, bare our innermost thoughts and feelings to determine
whether we truly share the same goals.”
“I’m being honest when I tell you that the Republic needs to be torn down and built up again from the ground up.”
“That is a tall order.”
“Tall, indeed.”
“It might require a civil war.”
“And how far from that are we now?” Dooku fell silent for a moment, then said, “The Senate grapples with trying to solve disputes the Jedi often see firsthand. What laws are enacted only follow from our having brought our lightsabers to bear.”
“It was the Jedi who pledged to support the Republic.”
“The Order’s place in this is a matter Sifo-Dyas and I have discussed endlessly,” Dooku snapped. “But the members of the Council are not similarly inclined. They are entrenched in archaic thinking, and slow to embrace change.”
He paused, and adopted a sinister look. “Don’t let yourself be fooled,
Palpatine. They see dark times ahead. In fact, they think of little
else. That’s why they have allowed the Jedi to become involved in
parochial conflicts like those at Galidraan, Yinchorr, and Baltizaar,
which are like brush fires born of windblown embers from a massive
blaze just beyond the horizon. But instead of actually rising up
against the corruption in the Republic, perhaps disbanding the Senate
entirely for a period of time, they have become fixated on prophecy.
They await the coming of a prophesized redeemer who will bring balance
to the Force and restore order.”
“A redeemer?” Palpatine stared at him in authentic surprise. “You’ve never alluded to this prophecy.”
“Nor would I now if I still thought of myself as loyal to the Order.”
“I never considered that the Force needed to be balanced.”
Dooku’s lip curled. “The Order interprets the prophecy to mean that the dark tide
needs to be stemmed.”
“You don’t accept it?”
Dooku had an answer ready. “Here is the truth of it: the Jedi could fulfill the prophecy on their own, if they were willing to unleash the full powers of the
Force.”
“The full powers of the Force,” Palpatine said. “I’m afraid
you’ve lost me.”
Dooku blew out his breath. “Perhaps it’s something we
can discuss in the future.”
“You’ve made your decision, then?” Dooku nodded. “If one more Jedi dies because of indolence on the part of the Republic and moral equivocation on the part of the Council, I will leave the Temple and refuse to look back.”
p. 293
[Palpatine] "...But if I am elected, and if you and I are going to begin to work together, it behooves us to give all appearances of being on opposite sides.”
Dooku folded his arms and stared. “Work together in what capacity?”
“That remains to be seen. But our common goal would be to return the Republic to what it once was by tearing it down.”
p. 344
[Dooku] “Because I suspect that Naboo was only the beginning—a kind of opening salvo. The Sith want to see the Republic brought down. Much as you and I do.”
Palpatine didn’t respond for a long moment. “But to ally with a Sith …”
“For many, they are the embodiment of pure evil, but the Council knows differently. What separates a Sith from a Jedi is the way each approaches the Force. The Jedi Order has placed limits on itself, but the Sith have never shied from incorporating the power of the dark side to accomplish their goals.”
“You wish to learn the secrets of the dark side?”
“I confess that I do.”
Palpatine restrained an impulse to reveal his true identity. Dooku was strong in the Force, and might simply be attempting to draw him out. On the other hand...
“Something tells me that this hidden Sith may eventually find his way to you,” he said at last. “And if and when he or she does, I hope that the alliance you forge will help us restore order to the galaxy.”
p. 365
