I know for a fact a computer must somehow add numbers but since I can only suggest it, it is forced to be a claim.
In that case, how could I ever state anything as a fact if I can only ever claim it?
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The relationship between these concepts is simple, though the devil is in the details. You can claim something is fact, but until it is proven, it is just a claim. What constitutes evidence and proof and whose burden it is is a major source of contention in philosophy. In a court of law, a judge declares something to be a fact, often with the aid of expert witnesses, such as a computer science. Often times, one claims something is a fact to either impose the view on a group of people or to share one's private thoughts on the actual nature of the claim; these are performativity and implicature, respectively.
The slightly circuitous route goes like this. In argumentation, we make utterances which are taken to be physical signals to others. An utterance is perceived to have a syntax which helps convey a meaning, that is semantic content. Semantic content in philosophy is known as a proposition. Sometimes two utterances with different syntax are the same proposition. For instance:
P1. I make an argument. (active voice)
P2. An argument is made by me. (passive voice)
These two expressions are largely taken to be synonymous.
Now, when an argument is had, there are essentially propositions that the parties accept, and those that can be debated. The former are known as facts, the latter as claims. What makes a claim a fact? Well, here's where things get sticky. For rationalists, rational deductive arguments. For empiricists, empirical evidence and inductive arguments largely. Most thinkers of course roll both approaches into their personal philosophy, and use what is known as inference to best explanation to boot, because good philosophers recognize that human reason is defeasible. When people agree on an approach to proof, the community has established what facts are and how they are demonstrated.
Well, according to social constructivists, essentially, 'facts' are claims about the agreement of other claims of truth, and according to John Searle in his Social Construction of Reality at least in the minds of the participants. Sometimes, when someone is making an explicit claim about a claim being a fact, they are using an aspect of language called performativity. So, when takes an oath to tell the truth and nothing but the truth, one is in a way performing to persuade others about the nature of the subsequent testimony. It may be that a proposition that makes claims about the factual nature of propositional content has a performative function. For instance:
Judge and Lawyer in camera discussing an appeal.
Lawyer (trying to pull a fast one): Your honor, that claim has yet to be proven!
Judge: I'm afraid that is not a fact. I am the trier of fact, and the claim the defendant raised is a fact because I said so. It has been proven beyond a reasonable doubt.
Lawyer: Oh, I see. I misunderstood.
Another motivation behind the claim might be to express an attitude about the nature of a claim. This is known as propositional attitude. If someone says:
It is a fact, that I exist.
The motivation for commenting on the claim may be to share their propositional attitude. Why is this necessary? Because from philosophy of language, sometimes there are attributes to an utterance that affect its meaning without affecting its syntax. Tone is a perfect example when one uses sarcasm. These subtle nuances are known in linguistics as implicature. From WP:
An implicature is something the speaker suggests or implies with an utterance, even though it is not literally expressed. Implicatures can aid in communicating more efficiently than by explicitly saying everything we want to communicate.
They can also be used to clarify what might be confused!
Ultimately, all of this parsing (can't help myself!) leads us to a sort of interpretation about truth and factuality that is usually classified under the idea of intersubjectivity.