In the video below, I'll share some basic tips and tricks for fastening these garments, but don't assume that way is the only way. The ancient Hellenes liked to spruce it up. Also, despite what the popular media says, ancient hellenic clothing was often made with brightly colored fabrics and beautiful trimmings; white most certainly wasn't the only color, nor the most preferred.
To recap: the peplos (ὁ πέπλος) was a body-length garment established as typical attire for women in ancient Hellas. It was a tubular cloth folded inside-out from the top about halfway down to form an apoptygma. The garment was then gathered about the waist and the open top--at the fold--pinned over the shoulders. The top of the tube was then draped over the waist, providing the appearance of a second piece of clothing.
The chiton (khitōn, χιτών) came in two forms: the Doric chiton and the later Ionic chiton. The Doric style was open on one side, sleeveless and didn't have the apoptygma the peplos had, while the Ionic chiton was pinned or sewn from top to bottom, was made with a much wider piece of fabric, and had sleeves.
A himation was a type of cloak that was usually worn over a chiton. It wasn't fastened with pins, but instead was held up by the friction created between two layers of the garment, usually over one shoulder. The chlamys (χλαμύς) was normally pinned with a fibula at the right shoulder. It was worn by messengers and warriors, and could serve as a light shield.
The peplos and chiton were usually made out of linen, silk, or other light materials. They were often tastefully translucent. The cloaks were usually made of wool. Women almost always wore a peplos or chiton under their cloaks (usually himations), but men were allowed to solely wear either of the cloaks. Women almost always wore ankle length (or longer) garments, while men were allowed to wear both ankle length as knee length versions.
I hope this video has provided you with a few tips and tricks. Good luck with making your own and enjoy!