When injecting PRF in the under eye area to rejuvenate, how many ml of PRF are recommended for each under eye area?

PRF stands for Platelet-rich fibrin, and differs from Platelet-rich Plasma (PRP), in the fact that the platelets are activated in PRF. Fibrin matrix is a kind of gel which acts to support cellular repair and rebuilding process. It's very sticky and difficult to inject through a needle, but there is a small window of time when it can still be injected, up to 15 minutes after the activation process starts. Activation could be through an external agent, such as a tiny amount of calcium chloride, or sometimes the natural clotting process if blood is collected without anticoagulant in the tube.
Both PRP and PRF can be good treatments for under eye problems, such as infraorbital dark circles, volume loss in the tear trough area, crepey skin, fine lines, etc. For more information including links to published research on this topic, please see the link included with this answer.
The amount injected would depend on a few factors: age of the patient, amount of volume loss, whether or not the upper cheek is being included (can be helpful as this area supports the under eye area). A typical amount (per side), if the cheek was not included would be 1-2 mL, and up to 5 mL if the upper cheek is included.
On average, patients can expect 2-3 days of swelling, but this can vary depending on all of the above factors, as well as the concentration of platelets and the total number of platelets injected. It's great if the treating clinic has access to a hematology analyzer to be able to give you these numbers, but this is not available everywhere.
Another thing to remember is that a series of treatments will usually give best results, for example 3 treatments spaced about 2-4 weeks apart.