Ok, so I watched the video you posted on my wall.
Things that I'm noticing:
1) You are playing extremely patient.
Now this isn't necessarily a bad thing, but it is costing you a lot of missed opportunities to capitalize on the punishes you hit on Peach. Peach does not enjoy getting hit by Fox (especially at low percentages due to his u-tilt). If you hit her, you need to continue your strings or follow ups. You are basically just playing a hit and run game which is good in certain instances. For example, when you are low percentages and there is a huge risk of her comboing you if you make a mistake. This is a good time to play that patient/burst playstyle so you can lower your chances of getting hit. However, once you reach like the 30%+ area, I really would suggest following up more on Peach, especially when she's at lower percentages. Nair will work really well against a grounded Peach, short hopped dairs also do the trick (but you can't be too obvious about them).
2) Too many shines.
It's alright to shine here and there, but you are just doing it randomly and repeatedly. Your opponent was picking up on it quite quickly, and it cost you damage as well as a death. You shouldn't use it anymore (especially on falling turnips), unless you know that your opponent is throwing a turnip at you in a horizontal direction or it helps you obtain a better position offstage.
3) Not enough pressure.
You need to pressure Peach a bit more. Fox in this matchup is actually considered "campy". I admire your short burst single lasers at lower percentages, but once you hit like 30%+ and don't run the risk of eating Peach's dair, I definitely recommend doing triple lasers or just grounded repeated lasers. Peach eats them quite quickly because her airdodge is so terrible and she's mostly approaching with a float. Moreso, you forcing Peach to approach rather than Peach forcing you to approach. Addition lasers will definitely handle that properly. Another thing I want to see is more dash attacks and grabs. Your opponent was grounded a lot and was throwing out f-smashes like crazy on both of y'alls first stock. If you would have shield grabbed / pivot grabbed it could have beaten it easily.
4) Go for more U-smashes
Attempt to read your opponent more often. Pick up on her float habits, as we as where she positions herself during her float. Fox's up-smash destroys Peach's positioning if you know how to space it correctly. Your opponent doesn't really seem afraid of it because you aren't throwing it out enough.
5) Abuse Peach In the Air
You were doing a fairly good job on this, but I really want to reiterate how much it can help you in the matchup. Full hop fair covers a ton of space, and has way too much priority for Peach to counter. Her airdodge is the absolute worst in the game and she's one of the floatiest characters to boot. If she's in the air you need to keep her up there by all means possible. Spaced nairs, fairs, uairs, and bairs can all help out a ton. Try doing some feint jumps to bait an airdodge, then going up there and doing whatever. If she's landing, just read her landing with an u-tilt or a shield grab.. one of the two will definitely work because the majority of Peach mains always airdodge upon landing.
6) Counterpicking
Against Peach, you do not want to take her to Battlefield, Yoshi's Island, Delfino, Rainbow Cruise, or Brinstar. All of these stages allow her to land easier on the huge platforms above her or they give you a disadvantage due to her air control.
You want to take her to Final Destination, Halberd, or Smashville. Halberd has low ceilings, so that's a given. Final Destination and Smashville have relatively no platforms for her to safely land upon which will aid you in keeping her up in the air.
Hope this helped, send more videos my way if you want additional critique.