Well, a lot of Zeldas here play her in different ways, so you might get various answers for this one. But here goes~
For me, you need to get in as her. That means landing one of our combo starters, those being Jab, Utilt, Dtilt, SH Nair, Landing Nair, or a grab into an Uthrow or Dthrow. Deslite what some may tell you, her combo game is actually pretty dang good. After that, try and extend your string as long as possible to get damage. It's important to keep on some pressure so the game doesn't go back to neutral, since Zelda's neutral is weak. It consists mostly of Jabs, Utilts, Dtilts, mid charge retreating Phantoms, spaced Fsmashes, and the occasional Din's Fire or Nayru's Love. You can also throw in some Nairs and LKs, but those are riskier options.
If you get them offstage, be aggressive. Zelda's offstage game is actually quite good with her Dair, Nair, Phantom, and amazing recovery. One Dair- even a sourspotted one -can end some characters (even more if you footstool right after). Dsmash is especially good as setting up edgeguards due to its angle.
For KOing, you want to try and KO them as early as possible. Killing power is one of Zeldas greatest strengths. She can Elevator at about 60-70%, and with rage, it can get into the 30s because Zelda is OP and needs to be banned (
). She can true combo into it too, so it's not a move only for hard reads and punishes. Then there's our Hoo Hah which can start KOing at about...105%. Baiting an airdodge after Dthrow can also set up a Bair Lighting Kick, which can KO anywhere from 60-100 depending on stage position. Usmash is another powerful KO option, killing Mario on FD at 105% (no rage).
Overall she wants to combo her opponents, harass them offstage, KO them early, and keep an early lead. Falling behind as Zelda is the one thing you absolutely do not want to happen.
Also just a personal opinion on Zelda here: She is much better than a lot of players give her credit for. She is still very flawed, but the amount of misconceptions about her is ridiculous. I feel like Zelda mains are the only people who actually understand how to play her, and even then some of them still don't make sense (coughGameFAQscough).
Sorry this got so long! But I hope this answers you question at least a little bit.