The word whether indicates that there is a choice or a doubt between two alternatives. After the word whether, the words or not are necessary in some cases, but optional in other cases, as in your two sentences.
The New York Times Manual of Style and Usage (Random House, 1999, p.355) says that “when a whether clause modifies a verb, or not is needed:
They’ll play tomorrow whether or not it rains.
(The clause [with whether] modifies ‘play.’)”
Other examples of sentences like this would be:
The project will continue whether or not the researchers receive a government grant.
The project will continue whether the researchers receive a government grant or not.
(The clause beginning with whether modifies “continue.”)
I’ll call you tomorrow whether or not I have the answer for you then.
I’ll call you tomorrow whether I have the answer for you then or not.
(The clause with whether modifies “call.”)
On the other hand, according to the Times Manual, or not may be omitted when the word whether introduces a noun clause used as the subject, object, or object of a preposition.
Remember that a noun clause functions as a noun and, theoretically, can have a pronoun or a noun phrase as a substitute: this, this fact, that, it, and this idea, for example. If it’s possible to make a substitution with a word like this or it or this idea, or not is not necessary to the sentence.
Here are some examples of sentences in which you may omit or not. Note that you can also include or not, and note, too, that or not can go at the end of the clause.
As subject of the sentence. “Whether (or not) he leaves” is the noun clause as subject. (You could also say: “This is of absolutely no importance to me.”)
(a) Whether or not he leaves is of absolutely no importance to me.
(b) Whether he leaves or not is or absolutely no importance to me.
(c) Whether he leaves or stays is of absolutely no importance to me.
If the words or not are omitted, there will probably be a statement of alternatives connected by or, as in this case “leaves or stays.” When the words or not are optional, their use emphasizes the idea that there is a choice.
As a noun clause as object of the verb. In the sentences below, wonder is the verb. “Whether (or not) he’ll get the job” is the noun clause. (You could also say: “Brad wonders this.”)
(d) Brad wonders whether he’ll get the job.
(e) Brad wonders whether or not he’ll get the job.
(f) Brad wonders whether he’ll get the job or not.
Informally, if is often substituted for whether, as in (d) and (f), but not (e).
As the object of the preposition. In the sentences below, on is the preposition. “Whether (or not) Alice wants to make the investment” is the noun clause. (You could also say: “The decision depends on this.”)
(g) The decision depends on whether Alice wants to make the investment.
(h) The decision depends on whether or not Alice wants to make the investment.
(i) The decision depends on whether Alice wants to make the investment or not.
If cannot be substituted in these sentences.
So, in answer to your question, with the example sentences
I asked him whether or not I should go.
I asked him whether I should go.
both whether and whether or not are fine, since they appear at the beginning of a noun clause used as the object of the verb asked. The entire noun clause—"whether (or not) I should go"—could be substituted with this: "I asked him this."