Why did Katniss agree to continue the Hunger games and kill madami children?

Because she's just a human being, not a flawless heroine or a saint. Her sister, the only person she was sure she loved, just died an awful death in front of her eyes. She survived two Hunger Games, her motherland was destroyed and her people were massacred, she's traumatized like hell, and she's only seventeen. It would be unrealistic if she didn't want revenge, I'd say.

When Katnis agrees to the hunger games, it is not for revenge. It was to keep on coins good side and having the excuse of prim does have coin and anyone of the matter to believe that Katnis's deep desire of revenge is present . So like stated before, coin has no reason to believe Katnis will try and take her out or rebel against coins actions. Katnis says in the book when it's tied 2 and 2 of mga boto and one madami needed to claim a yay or nay in the votes, "... And I can feel haymitch watching me.this is the moment, then. When we find out how much he truly understands me." And haymitch sagot with "im with the mocking jay.". Stating he knows she doesn't really want the hunger games again. But of her plan to take out coin easily.

She sinabi it's for Prim. So I take it she wanted revenge from the capitol.

She says that she wanted it for Prim, to avenge her death, but she might have just sinabi this because she was so upset about Prim's death that she wasn't thinking straight. Being in favour of it also kept her in Coin's good books which might have been part of the plan. But whatever reason she has for agreeing, she soon realises how wrong and hypocritical it would be and kills Coin when she realises she would rule in the same way as Snow.

She wanted revenge on the Capitol for Prim