Short Answer: Truthfully, we are not sure at all. We are actually pretty unsure of where/how the historical Jesus was executed and what happened after his death.
Long answer: In order to answer your specific question, we need to ask, if Jesus was crucified, what happened to his body? And, what do we historically know about the sequence of events that happened after his death.
This has been the subject of historiographical debates over the last two decades. Consensus wise, most Early Christian historians believe that Jesus of Nazareth was most likely executed by the state, and because the Roman state's typical form for punishment was crucifixion, this is most likely what happened to him. The area near Golgotha is a place known to have done execution, but so were other areas around Jerusalem.
Many historians lean on the side of John Dominic Crossan who has argued that in all likelihood he was executed and thrown into a mass grave. This is outlined in Bart Ehrman's How Jesus Became God: The Exaltation of a Jewish Preacher from Galilee and I think it was The Historical Jesus: The Life of a Mediterranean Jewish Peasant John Dominic Crossan's “The Dogs Beneath the Cross,” chap. 6 in Jesus: A Revolutionary Biography. The reason for our uncertainty is that secondary characters of antiquity are increasingly difficult to find out key aspects of their life, especially when their biographers were not witnesses and lived decades after the character's death.
Your question about the Tomb is even more difficult because the story as described in the Gospels is troublesome. None of them pass our tests for historical certainty, including the Criterion of dissimilarity and Criterion of multiple attestation. What do I mean by this? The stories don't actually match up. This is a problem for historians since every single resurrection narrative are completely different. A good example can be found if you ask what did they find at the tomb the morning of the third day? In Mark 16:5 it's One Young Man in Matthew 28:5 it's One Angel, in Luke 24:4 it's Two men, in John 20:12 - they don't find anyone there the first time they visit the tomb. This goes for every other detail in these narratives. Historians like Crossan and Ehrman both have also argued that the purpose of Roman Execution was to desecrate the body of the person being punished, thus allowing it it a burial in a tomb is highly unlikely (but not implausible).
So about that church in particular, no, the likelihood that he was executed and buried near that church is unlikely because we know nothing reliable about his death. It was picked up as the cites of both places for non historical reasons in the 4th century.
Edit: corrected a book title.
We can't really be sure.
Jerusalem was destroyed twice by the romans, in 70 by Titus and in 135 by Hadrian. After the second destruction the city rebuilt as roman city, renamed Aelia Capitolina and Jews were forbidden entry.
So any real information on the locations of the events of Jesus death was destroyed and built over.
Then in 326 arrived Helena, mother of the emperor Constantine, more than seventy years old. Constantine had recently unified the roman empire and legalized the Cristian religion. Helena was a devout Christian and started a program of excavations and she was said to have found the place of crucification, the tomb and so on (she was also supposed to have found the real cross, and plenty of other relics).
Of course many of those "finds" can be considered suspect; nearly three hundred years had passed from the events depicted in the gospels, Jerusalem had been razed and rebuilt at least twice in the meantime, and the local population had a clear incentive in helping the aged empress. Helena did also started the construction of many of the more famous churches, and shortly thereafter Jerusalem was the destination of lots of pilgrims.
We're not entirely sure a man named Jesus who said the things he is purported to have said in the Bible actually ever existed.
Outside of the Bible, historical evidence for his existence is scant, at best.