How do I make a circle with particles?

You could use this. That is a particle plotter by Cloud Wolf on YouTube that generates a function for Minecraft. The website has a video on how to use it (link is broken at the time of answer). To make the circle, in the drop down box choose sphere. Then, change the y value to 0. That will make a flat circle. To see it in the display press draw. If you want a flat circle on a different axis change the x or z value to 0 instead. For the animation you could make frames using schedule to run a separate function that runs the particle function at an entity you use to control position.

If you want to learn how to use the generator here is the link to the video about how to use the generator because as of writing this answer the button in the website doesn't work. If you have any questions please ask. I hope this helps.