How do you fat into meat when grilling or is there a better cooking method?

Fat comes out due to melting

The primary way that fat will leave a grilled meat is through melting. Animal fat isn't a pure or refined substance, so it doesn't have a set and exact melting point, the way that water does. In my experience, fat melting occurs in the 55 - 70 C range (130 - 160 F). You will see little fat gone at an internal temp of 55 C, but heavy melting at an internal temp above 70 C.

As an example, a rare steak at 52 C/125 F will still have most of its fat attached, although it will be softened (and delicious!). A pork butt cooked up to 90 C / 195 F will have almost all the fat melted; even if you left the thick fat-back on it. Therefore, fat retention is somewhat independent of the grilling method, it is all about the internal meat temperature. Cooking meat until "done" thus precludes leaving the fat in some cases. You don't want to serve your guests chicken or pork cooked only up to 55 C.

How to keep the fat in

If you want to keep fat in your meat even when cooked to high temperature, there are two solutions. First, you can leave a lot of fat on your meat in the first place. The second solution is to not let meats sit too long at high temperatures; after all just because fat is melted doesn't mean it immediately exits your steak, gravity has to pull it down and out.

An example of the first method is the aforementioned pork shoulder. The thick fat-back can be left up before cooking and, even over a 12 hour cook time, there will be plenty of fat left at the end. An example of the second method is a well marbled steak. The fat will start to melt, but the short cooking time at higher temperatures ensures that most of the fat stays where it is

On the other hand...

...maybe don't try to retain too much fat? Fat does add lots of flavors, but for items like chicken and certain cuts of pork, too much fat can be a bad thing. Also, an untrimmed steak can often include lots of connective tissue in the fat (depending on the cut) and that can make your steak gristly and unpalatable.

In general, I have never made any attempt to keep more fat in a grilled meat, of any cut. Much more often, I will trim to keep less fat. You need enough fat for flavor and juicy-ness, but too much often just ends up as a greasy meal.

I think that if you cook the edges of your meat very hot and fast, it will form a barrier that limits subsequent escape of fat during cooking. Here is what I do with burgers and steaks and bacon/turkey burgers:

1. My grill is gas. I get it as hot as it can get.

2. Meat goes on. Flip it after about a minute.

3. On flipping, heat turned down to as low as it will go.

The high heat sears the outside and locks in juice and fat. Subsequent cooler grill lets you cook it thru without overcooking or flaring up.

King is right about low and slow. The other way is to get a cut with lots of fat (pork shoulder or my favorite, ribeye roast) and cook it with the rotisserie and indirect heat. With the rotisserie the fat bastes the meat.