First, we read about the poison in Tyrion's trial:
Widow’s blood, this one is called, for the color. A cruel potion. It shuts down a man’s bladder and bowels, until he drowns in his own poisons. This wolfsbane, here basilisk venom, and this one the tears of Lys. Yes. I know them all. The Imp Tyrion Lannister stole them from my chambers, when he had me falsely imprisoned."
Why would we recieve a complete description about this one particular poison, and not the others, if we aren't to make use of this information? We are meant to understand the mechanics of widow's blood.
On the night Tyrion has Pycelle thrown in the Black Cells, he takes several jars from Pycelle's chambers.
When he was gone, Tyrion made a leisurely search of the quarters and collected a few more small jars from his shelves.
If Varys suspected that Tyrion had poisoned his sister, he would have realized right away that Tyrion had taken some bottles from Maester Pycelle's chambers, and would have the opportunity to blame anything that was missing on Tyrion, should it come to light. Pycelle would have blamed all of the missing bottles Tyrion, seeing his obvious opportunity. We don't know exactly when Varys (or a little bird sent by him) stole the widow's blood, but we can assume that Pycelle may not have realized until the trial.
Lord Tywin’s face was so dark that for half a heartbeat Tyrion wondered if he’d drunk some poisoned wine as well
This is during Tyrion's trial. This gives us the first real and explicit suggestion that Tywin might have been poisoned. A subliminal suggestion by G. Martin?
"Plan on a lengthy visit." Prince Oberyn sipped his wine. "You and Doran have many matters of mutual interest to discuss. Music, trade, history, wine, the dwarf’s penny... the laws of inheritance and succession. No doubt an uncle’s counsel would be of benefit to Queen Myrcella in the trying times ahead."
If Varys had his little birds listening, Oberyn was giving them a ripe earful.
"Your father," said Prince Oberyn, "may not live forever."
Something about the way he said it made the hairs on the back of Tyrion’s neck bristle. Suddenly he was mindful of Elia again, and all that Oberyn had said as they crossed the field of ashes. He wants the head that spoke the words, not just the hand that swung the sword. "It is not wise to speak such treasons in the Red Keep, my prince. The little birds are listening."
"Let them. Is it treason to say a man is mortal? Valar morghulis was how they said it in Valyria of old. All men must die. And the Doom came and proved it true."
This explicitly suggests that Varys' little birds are listening and would have gotten this information to Varys. If Oberyn had truly poisoned Tywin, I doubt he would have been so liberal in his conversation. Also, Oberyn knows, and admits as much, that any poisoning would sound suspiciously like the work of the Red Viper. It wouldn't be the brightest way for Oberyn to murder Tywin Lannister.
Tyrion walked slowly to the ladder, ran his hand across the lowest rung. "This will take me up to my bedchamber."
"Your lord father’s bedchamber now."
He looked up the shaft. "How far must I climb?"
"My lord, you are too weak for such follies, and there is besides no time. We must go."
"I have business above. How far?"
"Two hundred and thirty rungs, but whatever you intend-"
"Two hundred and thirty rungs, and then?"
"The tunnel to the left, but hear me -"
"How far along to the bedchamber?" Tyrion lifted a foot to the lowest rung of the ladder.
"No more than sixty feet. Keep one hand on the wall as you go. You will feel the doors. The bedchamber is the third." He sighed. "This is folly, my lord. Your brother has given you your life back. Would you cast it away, and mine with it?"
I don't believe that Varys expected or intended for Tyrion to kill his father, but seeing his chance to pin on another, he gave Tyrion all of the instructions. Varys knew he was going into hiding, so he wasn't worried about his own role in the poisoning. How did the poison get slipped into Tywin's drink? Why, a little bird, posing as a cup bearer.
Afterward he found Lord Tywin’s dagger on the bedside table and shoved it through his belt. A lion-headed mace, a poleaxe, and a crossbow had been hung on the walls. The poleaxe would be clumsy to wield inside a castle, and the mace was too high to reach, but a large wood-and-iron chest had been placed against the wall directly under the crossbow.
Varys knew that the crossbow was in easy reach; he had arranged for that furniture to be placed there. Why? That was his backup plan. A small child could easily reach the crossbow, kill Tywin while he was in bed with Shae, and sneak out through the hidden door. Or, if he needed to cast blame on the Tyrells, he may have come back after leading Tyrion away, and done the deed himself. He had already planted the Highgarden coin in Rugan's cell, remember. Both Tyrion's escape and the death of Tywin Lannister could have been pinned on the Tyrells.
He found his father where he knew he’d find him, seated in the dimness of the privy tower, bedrobe hiked up around his hips.
"Tyrion." If he was afraid, Tywin Lannister gave no hint of it. "Who released you from your cell?"
"I’d love to tell you, but I swore a holy oath."
"The eunuch," his father decided. "I’ll have his head for this. Is that my crossbow? Put it down."
Once Tyrion went missing, Tywin would have immediately suspected Varys, and so Varys would have had to kill him or else risk the strength and force of Tywin Lannister dragging him out for an inquisition.
Tyrion knew exactly where to find his father because he knew that he himself didn't steal Widow's blood, and that Varys' little birds were listening to Oberyn's damning speech and assumed that Varys had taken it upon himself to poison Tywin and blame it on Oberyn. It was easier for Tyrion to kill his father knowing that his father was already dead.
Ser Kevan's death is described in almost the exact same wording as Tywin's. Further reinforcing the idea that this was Varys' plan all along.
"Winter," said Ser Kevan. The word made a white mist in the air. He turned away from the
Then something slammed him in the chest between the ribs, hard as a giant’s fist. It drove
the breath from him and sent him lurching backwards. The white raven took to the air, its pale
wings slapping him about the head. Ser Kevan half-sat and half-fell onto the window seat.
What … who … A quarrel was sunk almost to the fletching in his chest. No. No, that was how
my brother died. Blood was seeping out around the shaft.
He stood in a pool of shadow by a bookcase, plump, pale-faced, round-shouldered,
clutching a crossbow in soft powdered hands. Silk slippers swaddled his feet.
The eunuch set the crossbow down.
...and using the exact same weapon.
you were threatening to undo all the queen’s good work, to reconcile Highgarden and Casterly
Rock, bind the Faith to your little king, unite the Seven Kingdoms under Tommen’s rule. So …"
A gust of wind blew up. Ser Kevan shivered violently.
We hear the rationale for Kevan's death, but wouldn't the same rational be used for Tywin's death since he also was in the process of reconciling the kingdom under Lannister and Tyrell rule?
Surely Varys had the means, the motive and certainly the opportunity to kill Tywin Lannister, and his death, whether by poison or by crossbow was already in the works before Tyrion got there. Tyrion just made everything that much easier for Varys.