All the sewing can be done by machine or by hand as you prefer. If you want to sew with historically accurate thread, you can use linen or wool thread, or silk if you prefer.
When done by hand, this can be done with a spaced backstitch, or a running stitch with the occasional backstitch.
On the machine a regular straight stitch in any size will work.
Felling the edges by hand can be done with a felling stitch, or a running stitch if you prefer. On the machine you can use any stitch you use to hem fabric.
When handsewing you can use a felling stitch or a running stitch. Or any other stitch you prefer for hemming.
On the machine, a straight stitch is a good option, but if you have a blind hem stitch that is an option too.
That's it you are all done! Now take a fibula (a brooch, pin or clasp) to fasten your cloak over your right shoulder (that is how the Romans did it, you can of course also take the left shoulder, or the middle, or whatever you prefer) and enjoy looking dramatic.