The kernel doesn't compile itself -- it's compiled by a C compiler in userspace. In most CPU architectures, the CPU has a number of bits in special registers that represent what privileges the code currently running has. In x86, these are the current privilege level bits (CPL) in the code segment (CS) register. If the CPL bits are 00, the code is said to be running in security ring 0, also known as kernel mode. If the CPL bits are 11, the code is said to be running in security ring 3, also known as user mode. The other two combinations, 01 and 10 (security rings 1 and 2 respectively) are seldom used.
The rules about what code can and can't do in user mode versus kernel mode are rather complicated, but suffice to say, user mode has severely reduced privileges.
Now, when people talk about the kernel of an operating system, they're referring to the portions of the OS's code that get to run in kernel mode with elevated privileges. Generally, the kernel authors try to keep the kernel as small as possible for security reasons, so that code which doesn't need extra privileges doesn't have them.
The C compiler is one example of such a program -- it doesn't need the extra privileges offered by kernel mode, so it runs in user mode, like most other programs.
In the case of Linux, the kernel consists of two parts: the source code of the kernel, and the compiled executable of the kernel. Any machine with a C compiler can compile the kernel from the source code into the binary image. The question, then, is what to do with that binary image.
When you install Linux on a new system, you're installing a precompiled binary image, usually from either physical media (such as a CD DVD) or from the network. The BIOS will load the (binary image of the) kernel's bootloader from the media or network, and then the bootloader will install the (binary image of the) kernel onto your hard disk. Then, when you reboot, the BIOS loads the kernel's bootloader from your hard disk, and the bootloader loads the kernel into memory, and you're off and running.
If you want to recompile your own kernel, that's a little trickier, but it can be done.