Can anyone please help me explaining the weight of the word claim in general purpose sentences?
For example, if I say
I will claim the amount.
Does it mean it is my right to get the amount, or it means that I may get the amount or may not?
There is no "general context" meaning of the word; the word "claim" can have one of many meanings depending on its context. As for your example:
I will claim the amount.
The word "amount" suggests to me that money is involved, and therefore I can assume that the meaning is this one (as per dictionary.com):
to demand by or as by virtue of a right; demand as a right or as due: to claim an estate by inheritance.
Thus, if you claim an amount, you declare that you have a right to that amount.
thefreedictionary.com: claim, definition 1: "To demand, ask for, or take as one's own or one's due"
To claim something does not mean that you will get it or even that you are likely to get it, just that you are asserting that it is rightfully yours. Whether you actually get it depends on whether others agree that your claim is valid, or you have the power to take it by force, etc.
I am reminded of an argument I saw on a web forum once. At one point one of them claimed that the other's latest argument was flawed and then said, "I declare myself the winner of the debate." The other replied, "Okay. I declare myself emperor of the world and order your immediate execution." Neither got what he wanted by simply claiming it.
The general meaning of claim is to state something to be true.
The claim that men are smarter than women is ridiculous.
This can be expanded, figuratively and implicitly to be stating that you have ownership of something.
I have claim [of ownership] on this land!
And by further extension, it can refer to the act of physically demonstrating said claim, as in your example:
He claimed his winnings.