I've used kinesiotape a good bit for various injuries. My own experiences appear to line up with the limited research that's been done on it: the benefits of it are mostly related to the tactile feedback on your skin. So to some extent it doesn't really matter if you use the "proper" taping method or not, as long as you have some tension on the skin, that will help.
It's costlier per foot than regular athletic tape, but it lasts a lot longer. Plus for most injuries you only need one strip of it. I have found that it works best for small aches and minor injuries to the knee, low back, ankle, hip flexor, IT band, and foot. I have not found it as helpful for calf, hamstring, glute, or other hip issues. It is NOT a miracle cure by any means. I think of it kind of like icing—a temporary way to relieve pain and allow you to get through training a bit easier.
One application of kinesiotape will last for several days at least. On dry areas of your body like your hip flexor or your knee, it will stay on for upwards of a week. More contoured areas like your Achilles, you're looking at more like 4 days. The one area that's a real pain to get it to stick to is your arch (for plantar fasciitis—but in that case you might as well use a low Dye taping with normal athletic tape anyways). Because your feet sweat so much it is tricky to get it to stay on. One trick I have used is the spray adhesive that trainers use for ankle tapings. If you have hairy legs, use a hair clipper to buzz off the hair over the area of the tape. It will last a lot longer.
Once you have applied the taping, YOU LEAVE IT ON UNTIL IT COMES OFF. This should be several days at least! Some edges will start to peel up, and you can trim them with a pair of scissors. You can shower, aquajog, etc., just fine and it should not come off. Just pat it dry with towel after.
You can get rolls on Amazon for not too much money. Let me reiterate that it is by no means a cure for anything. It just helps a little bit, and it's pretty convenient and doesn't interfere with your running like a normal athletic taping would, or even a reusable knee or ankle brace.
I've been meaning to put together a video on how I use kinesiotape. Maybe I'll do that this week. There are a few tricks to how to cut your strips and how to apply them that are hard to describe in words.