It takes exactly 1253 tries for Castiel to kill him. He knows; he counted each one.
The first one simply has to turn and smile at Cas to make him drop his angel blade, the clatter unnervingly loud against the floor. When Naomi steps forward to take care of it herself, Castiel whirls and snarls at her "You will not touch him!" She still guts the copy of Dean anyway, and leaves Castiel to come to terms with the fact that she will not rest until he's able to kill his best friend.
The third one seems startled when he sees Castiel come up to him, trembling strongly but his hand gripped around his sword. "Cas?" He asks in confusion, and that's all it takes for the angel to practically collapse, murmuring "I'm sorry, I'm sorry, I'm so sorry..." Naomi isn't sure whether he's speaking to her, or the copy of Dean.
By Dean 32, Naomi becomes frustrated and stabs the copy herself. Castiel lunges at her, blinded by his fury, but she only needs to hold off his attacks a little while until a few of her subordinates come and take him away. Before leaving herself, Naomi stops by the dead body and curls her lip at it. "What is it about you-" She starts, but shakes her head and leaves the room, heels clacking loudly in the silence.
Number 165 is the closest Castiel has gotten to actually killing the copy himself. He grabs Dean by the collar of his shirt and backs him into the wall, breaths coming out in erratic gasps. But Dean simply looks at him, confused for a moment, before a sly smile works its way onto his face, and he leans forward. Castiel is caught off-guard by the kiss, but in a heartbeat angles his head to deepen it, hands clutching desperately at Dean's leather jacket. With an angry growl, Naomi has another angel haul Castiel off the copy and stabs it viciously, ignoring the pained cry that comes from behind her as Dean collapses.
Now, after each failed attempt, Naomi 'disciplines' Castiel. She's starting to wonder if she'll ever get him to kill the Righteous Man.
Attempt 475: Castiel stabs Dean this time. Only enough to wound, but it's progress. As a reward, Naomi leaves him to weep into the copy's shoulder, pretending she doesn't see how 'Dean' cards his fingers through his hair, murmuring softly. Later on, Naomi disposes of it in front of Castiel, showing him that his work is not done.
By copy 597, Castiel can give Dean a fatal wound. He never finishes the job himself, unable to give an actual killing blow, but the copies die anyway in his arms. Now, they murmur to him that they love him, that they cannot understand why he's doing this, because they're family. All it does is make Castiel cry harder.
Dean number 734 is what makes Naomi change her mind about her tactics. The other copies... They spend too much time grovelling. So this time, Naomi has 'Dean' spit words of hatred at Castiel as he stabs. This time, he actually manages a killing blow, struggling to keep his face a stony mask as he kneels besides the corpse. Even though he doesn't sob openly, Naomi can see the tears drip down his face, and the slight trembling in his shoulders. She smirks discretely. It took time, but he's finally getting there.
"I trusted you!" Dean #938 snarls angrily at Castiel. "I thought that... That we-" Before he can finish his sentence, Castiel stabs him in the heart. The light fades from his green eyes and he slumps forward against Castiel. The angel takes in a shaky breath and keeps his arms wrapped around Dean, but his face is impassive. Naomi comes up from behind and rests a hand on his shoulder. "Good, Castiel." She says, pleased. "You're almost there." Castiel leans his forehead against Dean's shoulder, but doesn't say anything to disagree.
Copy 1076 goes back to using Castiel's emotions against him. It still doesn't stop the angel from slitting his throat, but Naomi catches a glimmer of pain in his deep blue eyes as he stares down at the corpse. She doesn't think much of it; it'll be driven out of him with enough practice.
Finally, finally, Naomi deems Castiel ready to take down Dean. He executes copy 1253 with perfect precision, and there isn't a hint of remorse in his eyes when Dean begs him to stop. When he's dead, Naomi smiles. "You're ready, Castiel." She says softly, eyes fixed on the corpse. "At last, you're ready."