Only a jury can answer your question, and then on a case-by-case basis.
When a witness testifies, the question of whether and to what extent to believe/disbelieve a witness is nearly 100% the decision of the jury (or the judge in a bench trial).
In a criminal trial, at least, if a thousand disinterested nuns can take the stand and give identical testimony that Defendant robbed the bank, and Defendant's mother says he was actually with her on a Bahamas cruise at the time of the robbery (nevermind the fact that he was arrested exiting the bank), the jury is 100 percent free to conclude the nuns are lying and the mother is telling the truth.
Likewise, if Superman is on trial, the jury would be free to believe or disbelieve whomever they chose if the only evidence they had to rely on was (1) Lex Luthor's testimony that he hates Superman and saw him kill Doctor Light; and (2) Lois Lane's testimony that she loves Superman and saw Lex Luthor do it.
The jury's right to weigh credibility is hundreds of years old.
The question of how to weigh this evidence -- and the answer, which is that it is essentially left entirely to the jury -- is among the most well-settled principles of common-law evidence. See, e.g., Blackstone, Commentaries, 3:354 (1768) ("All others are competent witnesses; though the jury from other circumstances will judge of their credibility.").
Jury instructions therefore routinely advise jurors that they may consider a witness's prejudices for or against the defendant and conclude that his statements are not reliable, whether because he is lying or simply too biased to reliably remember the events in question. United States v. Abel, 469 U.S. 45, 52, 105 S. Ct. 465, 469 (1984) ("Bias is a term used in the 'common law of evidence' to describe the relationship between a party and a witness which might lead the witness to slant, unconsciously or otherwise, his testimony in favor of or against a party. Bias may be induced by a witness' like, dislike, or fear of a party, or by the witness' self-interest.").
You can follow the trail of cases supporting this position quite a ways back. I stopped when I reached Honegger v. Wettstein, 94 N.Y. 252 (1883), a case where a manufacturer sued members of a firm that had bought its goods but never paid. At trial, a principal of the firm testified about the invoiced amount to be paid, and because the testimony was uncontradicted, the judge ordered a directed verdict in favor of the buyers. But the New York Court of Appeals reversed, holding that the credibility of a biased witness's testimony must be left to the jury:
Although not contradicted, he was an interested party, and had a direct interest in increasing the fund in the hands of the receiver, and in preventing its payment to the plaintiffs. His evidence was given for the purpose of showing the alleged violation of law by the plaintiffs, and in explanation of the three invoices which were made of the goods, of which duplicates were made, and it was a fair question for the jury to say whether he might not have been influenced by the circumstances stated.
Ten years later, the Supreme Court addressed your question a bit more directly. In Reagan v. United States, 157 U.S. 301 (1895), a defendant who took the stand was convicted after the trial court told the jury that "Where the witness has a direct personal interest in the result of the suit the temptation is strong to color, pervert, or withhold the facts." But the Supreme Court affirmed the conviction, holding that "the court may, and sometimes ought, to remind the jury that interest creates a motive for false testimony."
Modern courts permit, but do not require, instructions highlighting a specific witness's bias.
These cases of course deal with witnesses who have a bias in favor of a party, but the rule is no different in cases where a witness, as in your question, admits to a bias against a party. In United States v. Coleman, 887 F.2d 266 (6th Cir. 1989), for instance, a defendant was charged with conspiracy to commit arson. Because the only direct evidence of his agreement to the arson was testimony from a witness "who admitted a bias against defendant," he sought a judgment of acquittal, but the trial court denied his motion and sent the case to a jury. The Sixth Circuit affirmed his conviction, holding that it was up to the jury to decide whether the witness was credible:
As defendant points out on appeal, Boscaglia's credibility was very much in issue at trial, but the jury resolved the issue against defendant. The jury's decision on "the credibility of witnesses is not reviewable on appeal."
Although such instructions are allowed, this does not mean that a court is always required to give an instruction addressing the credibility of a specific witness. Depending on who the witness is biased for/against, there are likewise cases saying that no instruction was necessarily required, Kovacich v. Spearman, No. 2:13-cv-0985 KJM DAD P, 2015 U.S. Dist. LEXIS 108233 (E.D. Cal. Aug. 14, 2015) (denying habeas claim by defendant who unsuccessfully sought jury instructions highlighting the bias of the investigating officer), or that such an instruction is a bad idea but not disallowed, United States v. Jones, 372 F. App'x 343, 345 (3d Cir. 2010) ("While such an instruction that singles out the defendant's interest in the case is not advisable, we find that here it did not constitute reversible error.").
I haven't found any saying that such an instruction was a reversible error, though I suspect that there are cases out there in which verdicts were reversed because the judge went overboard in suggesting to a jury how much credibility they should extend an admittedly biased witness.
