Yes, Epstein killed himself.
The NYC medical examiner ruled it a suicide: https://www.nytimes.com/2019/08/16/nyregion/jeffrey-epstein-autopsy-results.html
Attorney general William Barr called it a "perfect storm of screw-ups" after being initially suspicious that it might have been a murder: https://www.nytimes.com/2019/11/22/nyregion/william-barr-jeffrey-epstein-suicide-investigation.html
DoJ and FBI investigation into Epstein's death resulted in 2 prison guards being indicted for falsifying records - basically that they said they checked on Epstein regularly while instead they took naps and shopped online: https://www.nytimes.com/2019/11/19/nyregion/epstein-prison-guards-arrested.html
Probably the best source of what happened is the indictment of Michael Thomas and Tova Noel, the prison guards on duty the night Epstein died. Here are select parts:
The full indictment can be found here:
Allegedly there is a video of the night in question showing no one entered the unit, but I've not been able to find that yet. Hopefully, it will be released at some point.
To summarize: Epstein was put in a single cell, the guards stayed outside of the unit, and, allegedly, video footage shows nobody entered the tier where Epstein was housed. Then at 6:30 am the guards entered the tier, found Epstein dead with a noose around his neck, and triggered the alarm at 6:33 am.
Given the evidence as it's known now, this is clearly a suicide.
Edit: The prison guards have admitted to falsifying records and were sentenced to probation. The charges against them were ultimately dismissed
Tova Noel and Michael Thomas admitted that they “willfully and knowingly” lied on forms stating that they’d made the required rounds checking on inmates the night of Epstein’s August 2019 suicide
A Manhattan judge has ordered charges dropped against the two prison guards who admitted to falsifying records after Jeffrey Epstein committed suicide in jail more than two years ago.
The guards, Tova Noel and Michael Thomas, copped prosecution deals in May that required them to admit their guilt, with the understanding that the charges against them would be dismissed if they followed the rules of their agreement for six months.
The pair also had to carry out 100 hours of community service as part of the deal.
Update - June 2023:
DOJ office of inspector general just published a report concluding that Epstein death was indeed a suicide:
No Evidence Contradicting the FBI’s Determination that there Was No
Criminality Associated with Epstein’s Death. Separate from the OIG’s
investigation, which focused on the conduct of BOP personnel, the FBI
concurrently investigated whether Epstein’s death was the result of
criminal conduct by any non-BOP actors. Among other things, the FBI
investigated the cause of Epstein’s death and determined it was not
the result of a criminal act. The Office of the Chief Medical
Examiner, City of New York, determined that Epstein died by suicide.
While the OIG determined MCC New York staff engaged in significant
misconduct and dereliction of their duties, we did not uncover
evidence contradicting the FBI’s determination regarding the absence
of criminality in connection with Epstein’s death.