Here's my understanding/theory of his betrayal. I apologize for the long post.
It is shown in the beginning of The Introduction that earlier in 1992, while CJ was still living in Liberty City, Officers Frank Tenpenny and Eddie Pulaski went to talk to Big Smoke. They were not at his house in Idlewood however, but at Ganton Courts across from the Ganton Gym (see photo).
After coming out of one of the apartment buildings (seemingly the one shown in the above image), Tenpenny said "I knew that fat f*ck would see it our way," to which Pulaski replied "always do, once they see the choices we're offering." Pulaski then asked Tenpenny what they were going to do about Officer Hernandez joining their patrol with them. Tenpenny replied "exactly the same thing as last time; he's either going to play our way, or he's gonna have a problem like that oversized asshole back there," referring to Big Smoke.
This suggests that Big Smoke was coerced by Officers Tenpenny and Pulaski, and that he reluctantly agreed to whatever they were ordering him to do.
The next morning, Big Smoke went to Ryder's house on Grove street, and the two were seen already talking in the kitchen. Ryder said "the way I see it man, I need to be in charge of my destiny homie." Smoke replied "this way, you will be homie. This is all about destiny." So Big Smoke was coerced by Tenpenny and Pulaski, and brought it up to Ryder, who seemed to see it the same way as Smoke.
With Ryder now on board, Smoke continued: "now the way I see it, we don't have a choice. Ain't nothing to feel bad about; you put a gun to a brotha's head, brotha's gonna do what he's told, no matter if he's a fool or a wise man. You don't want a bullet in your brain. But if you can make it work, and make some benjamins whil you're at it, [laughs] then that's cool." So the night before, Officers Tenpenny and Pulaski forced Big Smoke's hand about something, to which he was reluctant enough about to have one, or both officers put a gun to his head. Before going to talk to Ryder about it, he thought long and hard about what he was being forced to do, and though he started out reluctant, he soon began to see an opportunity -- his own rise to power, even while under duress.
A few days later, a game of dice was being played in the cul-de-sac on Grove Street, with Big Smoke, Sweet, and a few unnamed Grove Street gangmembers, as well as OG Loc. Big Smoke said to Sweet "hey fool, we're losing the streets, man." Sweet replied "nah, we're just standing by our principles, homie," referring to Sweet's (as the leader of the Grove Street Families) resolute refusal to involve his gang with the drug scene going on in Los Santos. Big Smoke argued his case: "but our principles are making us bitches, man! Every day, Ballas get stronger, and you and I get weaker; you and I get poorer man!" Sweet simply replied that it would all blow over; "it always does."
So even though Big Smoke had begun his betrayal under duress, and later accepted it after forseeing his rise to power, he still tried to plead his case to Sweet; that Grove Street's only way of becoming the strongest gang in Los Santos again was to enter the drug dealing scene, which Sweet refused, staying true to his and his gang's principles. I believe that it was at that moment that Big Smoke's betrayal of Sweet and the Grove Street Families was cemented -- Big Smoke was ordered by Officer Tenpenny to organize a hit on Sweet to remove him from power. With Sweet then dead, Big Smoke could either disband the GSF or involve them in the drug scene, as well as Tenpenny ensuring Big Smoke's ascension to power as the leader of the drug operation in Los Santos that was making Tenpenny loads of money through extortion, and which would also financially secure Big Smoke, making him a household name on the Los Santos streets. In betraying the Families, Smoke would also have to secretly align with the Ballas, the Families' arch rivals, and the gang that Officer Tenpenny was also allied with.
Several days later, Officer Tenpenny is seen in Market, Los Santos, talking on the phone to Big Smoke. Tenpenny says "listen to me, son: I don't give a f*ck about you, I don't give a f*ck about your principles, I don't give a f*ck about your friends. People who get in my way get f*cked with. Now, you got paid. You took the money." So Big Smoke had already been paid for organizing the hit on Sweet. Tenpenny continued "I'm trying to set you up for life here, boy, and you're inadequate...do what we agreed...this week!"
At some point that week, Big Smoke did what he agreed to do: organized the hit on Sweet. Late one night, a Green Sabre with two Balla gangmembers (who Big Smoke had secretly aligned with without Sweet's knowledge) drove down Grove Street, and into the cul-de-sac. As they approached Beverly Johnson's house, the Balla in the passenger's seat opened fire with an automatic weapon, hitting the house with bullets, intending to kill Sweet, who was believed to be inside the house. He was in fact across the cul-de-sac at the time of the shooting, and CJ's and his mother Beverly was killed instead. The car tore off into the night, with Sweet and his sister Kendl running into the house to find their mother had been killed in the shooting.
Sweet then called CJ, who was living in Liberty City at the time, and informed them of their mother's death. A few days later, CJ arrived in Los Santos.
Another thing to note: at the end of the mission Drive Thru, Big Smoke asks you to take him home. You take him to his house in Idlewood (not Ganton Courts, where he was presumably living when Tenpenny and Pulaski went to talk to him at the beginning of The Introduction. CJ asks him why he "moved out the Grove." Big Smoke says that he "got some money from his aunt." I believe that to be a blatant lie, and that he bought the house with the money he received from Officer Tenpenny for organizing the failed murder of Sweet during The Introduction.