How to read / write .exe machine code manually?

The executable file you see is Microsofts PE (Portable Executable) format. It is essentially a container, which holds some operating system specific data about a program and the program data itself split into several sections. For example code, resources, static data are stored in seperate sections.

The format of the section depends on what is in it. The code section holds the machine code according to the executable target architecture. In the most common cases this is Intel x86 or AMD-64 (same as EM64T) for Microsoft PE binaries. The format of the machine code is CISC and originates back to the 8086 and earlier. The important aspect of CISC is that its instruction size is not constant, you have to start reading at the right place to get something valuable out of it. Intel publishes good manuals on the x86/x64 instruction set.

You can use a disassembler to view the machine code directly. In combination with the manuals you can guess the source code most of the time.

And then there's MSIL EXE: The .NET executables holding Microsofts Intermediate Language, these do not contain machine specific code, but .NET CIL code. The specifications for that are available online at the ECMA.

These can be viewed with a tool such as Reflector.

OllyDbg is an awesome tool that disassembles an EXE into readable instructions and allows you to execute the instructions one-by-one. It also tells you what API functions the program uses and if possible, the arguments that it provides (as long as the arguments are found on the stack).

Generally speaking, CPU instructions are of variable length, some are one byte, others are two, some three, some four etc. It mostly depends on the kind of data that the instruction expects. Some instructions are generalised, like "mov" which tells the CPU to move data from a CPU register to a place in memory, or vice versa. In reality, there are many different "mov" instructions, ones for handling 8-bit, 16-bit, 32-bit data, ones for moving data from different registers and so on.

You could pick up Dr. Paul Carter's PC Assembly Language Tutorial which is a free entry level book that talks about assembly and how the Intel 386 CPU operates. Most of it is applicable even to modern day consumer Intel CPUs.

The EXE format is specific to Windows. The entry-point (i.e. the first executable instruction) is usually found at the same place within the EXE file. It's all kind of difficult to explain all at once, but the resources I've provided should help cure at least some of your curiosity! :)

You need a disassembler which will turn the machine code into assembly language. This Wikipedia link describes the process and provides links to free disassemblers. Of course, as you say you don't understand assembly language, this may not be very informative - what exactly are you trying to do here?