Effectively diagnosing a computer, properly and efficiently in other words, means first understanding how a computer works. It's a way of thinking. Some people have it, many don't. It's a "logical/analytical" thought process. Quite literally, a flow chart.
The reason I'm good at what I do is because I think like a computer. For every problem, there's a logical explanation (when something illogical happens, which is fortunately very rare, it really screws me up. lol)
That's why I can come in here and bypass most steps others have to take in order to figure out the hard drive is in trouble, for instance.
Anyways. Once you know how the computer works, how all the parts work together, the flow of information, etc.. When something stops happening you can backtrack. You ask yourself "what needs to happen in order for THIS to work properly?" and then you use the process of elimination. In most cases I can diagnose a computer in under five minutes.
I can't really sit here and write a whole tutorial on how a computer works.. I simply don't have the "patience" or "focus"... But what I can tell you is that you can eliminate the obvious.. RAM, hard drive, optical drive, display, voltages, etc. What you want to do is look into a tutorial on what happens when a laptop posts. Ask yourself "what's going on?". "What processes happen at this point and when they don't, why don't they?"
Once you've done that, know that about one in every 5000 CPUs will go bad.. A BIOS will corrupt, preventing the board from posting, LONG before most CPUs will go belly up. You also get to know which problems are most common with which computers. If you had a week or two I could list off many of them, but you'll see it pop up in my posts (hint: WD Caviar SEs).
With the HP/Compaq, I know that the AMD versions with the nVIDIA chipsets go bad on a regular basis. So much so that the only laptop board I stock on a regular basis is the different models of DV series nVIDIA based boards.