The claim is very old and very specific: that the high altar of Saint Peter's stands directly above the grave of Peter. It has been tested more thoroughly than almost any comparable claim in Christianity, and the result is genuinely mixed.
Why the building is where it is
Constantine's basilica, begun in the 320s, was built in an extraordinarily awkward place. The Vatican hill had to be cut into on one side and built up on the other; a working cemetery, still in use and legally protected, had to be filled in and buried. Level ground was available a short distance away.
No one has produced a satisfactory explanation for that expense other than the obvious one: the builders were fixing the position of the altar over a spot they were not free to move.
What the excavation found
Excavations ordered by Pius XII ran from 1940 to 1949, in secrecy, beneath the basilica floor. They uncovered a Roman street of mausolea, mostly pagan, some later Christian, running under the nave. Directly beneath the high altar they found a small structure of about the year 160: a niche with two columns set against a red wall, matching the description by the presbyter Gaius, writing around the year 200, of the tropaion or monument of Peter on the Vatican.
That much is agreed. A second-century Christian monument, marked as Peter's, in the location the tradition specifies, under the altar. It is a striking result and it is not seriously contested.
The bones
The contested part is what was in it.
Bones found in the immediate area during the excavation were initially set aside and, on examination, included animal remains and the bones of more than one individual, including a woman, which was reported and effectively closed the question at the time.
Margherita Guarducci, an epigrapher, later argued that the relevant bones had been removed from a marble-lined repository in the adjacent graffiti wall and stored separately without being catalogued. She read the wall's dense Greek graffiti as containing the phrase Petros eni, "Peter is within", and the bones she identified were of a single robust male of about sixty to seventy years, wrapped in a cloth of gold and purple.
Paul VI announced in 1968 that the relics of Peter had been identified.
How to hold it
The disagreement is real and it runs along a fault that is not simply belief against scepticism, some of the sharpest criticism has come from Catholic archaeologists, including members of the original excavation team, who objected to the chain of custody, to the fact that the bones were not recorded in situ, and to the reading of the graffiti.
The honest summary: the shrine is second-century, is certainly Petrine in intent, and stands where the tradition says it stands. Whether the specific bones announced in 1968 are Peter's rests on a reconstruction after the fact, and is not established by the standards that would be applied to any other burial.
The basilica is meanwhile a working church rather than an archaeological site. The necropolis is open only to small booked groups, partly for conservation and partly because it is directly beneath a functioning altar.