When refurbing my celestion hi-fi speakers many years back, I struggled to get the cloth to a good tension and wrinkle free.
A good little trick when doing grill cloth, is get it neatly around the frame just shy of the tension you want/need then hit it with a heat gun around the edges.

You want a medium heat and not too close to the cloth. I recommend experimenting on some offcuts of cloth to find the right combination of heat & distance. IT WILL MELT if too hot, too close. You just want to "shrink" it a bit to impart tension.
Work the four edges first, then methodically work you r way into the middle.

By doing the heat gun trick you don't have to fight with the cloth so hard with trying simultaneously stretch it, hold it and staple it.