A few months ago I found Link to be a hard matchup for me. It's getting easier since I play with a very good Link main in my area. Ok time for advice.
You have to remember that Link is using his projectiles to keep you away because he know that your close range game is better overall. Use your turnips to close in the distance to the Link player and force him to come with in your comfort range: close range. Floating Canceling (FC) your Bair and Nair are good ways at closing in the distance to the Link man. They have high priority and it should stop the boomerang if timed correctly.
The FC Bair or Nair should be pretty close to the Link as well and it limit his options especially if you land this attack or hit his shield. If Link attempts to shield the FC Bair or Nair, aim for a grab, dtilt, jab or dash behind him. Link will have enough shield stun for you to do most of Peach's moves safely once you are in close. Good Link will attempt to Nair out of shield, Up B out of shield or just wait and roll out of shield. Pay close attention to Link's Shield game and punish accordingly and quickly. Keep up the pressure!
Long range is where you dont want to be and Link will rack up the damage and attempt to capitalize on your approach weaknesses. Do not Dash Attack a lot against Link at this range. It's a free shield grab for Link. I remember a year ago when a played Kubuu and making this mistake. Link has nasty grab combos on Peach and you want to reduce the amount of grab opportunities of Link. Instead float midrange in the air and force him to whiff a boomerang toss. Land Fairs and Dairs while closing in the distance. The Link will either attempt to WD Back and Fsmash or perform a retreating Fair if you miss these attacks; however, you should be able to shield out of the FC and you'll be close to Link which is the range you want. WD out of shield after the FC will also help you close the distance on Link as well. You can even perform a quick dash attack out of the FC as well if you expect a range and you'll surprise him good (best way to setup up Dash Attacks on Link)
This is one of my best matchups and I find myself doing extremely well against most Links. Tell me more about his habits and I'll provide some insight on what I do in that instance.
-Peach Out!