An LKM lives in a single ELF object file (normally named like "serial.o"). You typically keep all your LKM object files in a particular directory (near your base kernel image makes sense). When you use the insmod program to insert an LKM into the kernel, you give the name of that object file.
For the LKMs that are part of Linux, you build them as part of the same kernel build process that generates the base kernel image. See the README file in the Linux source tree. In short, after you make the base kernel image with a command such as make zImage, you will make all the LKMs with the command
This results in a bunch of LKM object files (*.o) throughout the Linux source tree. (In older versions of Linux, there would be symbolic links in the modules directory of the Linux source tree pointing to all those LKM object files). These LKMs are ready to load, but you probably want to install them in some appropriate directory. The conventional place is described in Section 5.5. The command make modules_install will copy them all over to the conventional locations.
Part of configuring the Linux kernel (at build time) is choosing which parts of the kernel to bind into the base kernel and which parts to generate as separate LKMs. In the basic question-and-answer configuration (make config), you are asked, for each optional part of the kernel, whether you want it bound into the kernel (a "Y" response), created as an LKM (an "M" response), or just skipped completely (an "N" response). Other configuration methods are similar.
As explained in Section 2.3, you should have only the bare minimum bound into the base kernel. And only skip completely the parts that you're sure you'll never want. There is very little to lose by building an LKM that you won't use. Some compile time, some disk space, some chance of a problem in the code killing the kernel build. That's it.
As part of the configuration dialog you also must choose whether to use symbol versioning or not. This choice affects building both the base kernel and the LKMs and it is crucial you get it right. See Section 6.
LKMs that are not part of Linux (i.e. not distributed with the Linux kernel) have their own build procedures which I will not cover. The goal of any such procedure, though, is always to end up with an ELF object file.
You don't necessarily have to rebuild all your LKMs and your base kernel image at the same time (e.g. you could build just the base kernel and use LKMs you built earlier with it) but it is always a good idea. See Section 6.