Q: How can I reach points???

If you mean get points on these forums then you need to reply to other people's questions, and they have to mark your reply (you can't ask for points) as having helped (5 points) or solved (10 points) their problem. There is more info here : https://discussions.apple.com/docs/DOC-9251

Other forum tutorials : https://discussions.apple.com/community/tutorials

By answering other people's questions.

Award points, level up, and earn new privileges - https://discussions.apple.com/docs/DOC-9251

You acquire points by answering questions. If the poster feels that your answer is helpful or that it solves their issue, they can mark their question as such. Also, if Apple feels that a particular answer is valuable for future posters, they can mark it as Apple Recommended. You get 5 points for Helpful, 10 points for Solved, and 7 points for Apple Recommended. You are not allowed to ask for points in a response, however.

Cheers,

GB

How long does it take before the points show up?

I received an e-mail on 2 Oct 2016  stating I would receive 10 points for Solving an issue but still don't see the points.

Do I have to register somewhere to apply for the points?

If you are referring to this thread : Upgrade from OS X 10.7.5 Lion to 10.8.5 Mountain Lion

then you won't get any points as you started the thread, you will only get points from answering other people's threads and your replies being marked as having helped or solved their question (only the thread starter can do that), or the community specialists on here marking your reply as 'Apple recommended'.

Thanks - that clears it up (although in the thread, I solved anneo_bettyb's problem, who jumped into the thread after I had started it). 

It doesn't matter if you were replying to somebody else in the thread and solved their problem, it's who started the actual thread that's important.

There is another curiosity too.   You seem to have awarded three lots of help points (5 points each to three people).   According to the rules only two helps should have gone through, however many times you pressed the button.   But everyone has given you right answer so I guess all is well anyway.  :-)