How can I reset all my TF2 achievements?

I don't believe there's a completely legit way to do this.

However...

If you're willing to risk a VAC ban (however unlikely it may be), look at the Steam Achievement Manager. Sketchy-looking site, people use the tool to cheat, use at your own risk, etc. etc. etc. Here are some old Steam forums of people talking about SAM, as well as a previous StackExchange post on the topic. It's against the rules, it's a moral grey area, but for your described purposes, I personally don't see an issue with it.

Otherwise, create a new Steam account, spend $5 on something so you can download all the free games, and start fresh

As of June 7 2018, you can reset achievements in Team Fortress 2 from the "View Achievements" menu.

• Added "Reset All Achievements" button to the Achievements dialog
  • A small number of achievements are based on player stats and will be awarded again when Team Fortress 2 is restarted unless the player stats are also reset
  • Resetting achievements will not allow you to receive achievement-based items multiple times

Team Fortress 2 Patch Notes - June 7, 2018

Try: Launch TF2, click "items", select the stats tab, and then select "reset stats" to the bottom right. Your loadouts will be reset however, and you'll have to re-add the items onto your classes. (Source)

Alternately, send Valve a support ticket asking to have your achievements for TF2 reset, but expect to wait a week or so. (Source)

Note: Achieving previously achieved achievements again will not result in you getting the achievement items. (Source)