I remember Journey to the Center of Earth and wonder: What is the deepest in the Earth surface or below sea level we have traveled either by foot, sub, drill or camera?
Probably a bit over 4 km, in this South African mine: https://en.wikipedia.org/wiki/Mponeng_Gold_Mine But as the link mentions, the mine operators go to considerable lengths to reduce the mine temperature to endurable levels from the 66°C/151°F of the surrounding rock.
Note: This answer is for the original question, where the OP asked for the deepest depth below the surface. It's since been changed to ask for depth below sea level.
Since you termed it based on sea level, the gold mines in South Africa are not the deepest, they begin at an elevation of ~1500 m, meaning their 4 km depth is only 2.5 km below sea level.
The Kidd mine in Canada is 2.9 km deep and is located at an elevation of only ~250 m above sea level making it's depth 2.65 km below sea level.
Also if you don't care about the underground part and only care about below sea level, the undersea exploration of Challenger Deep wins by a lot, they went ~12 km below sea level.
You generated multiple questions, I am going to list some of them. People can disagree with any of my answers, I made the estimations quickly.
1.Record depth in the rocks from the surface a person has reached.
Based on your original question, jamesqf's answer of the mine in South Africa may be the deepest.
2.Record depth in the rocks from the surface a person has reached -see level is the reference-.
Based upon "under sea level", Josh King's answer of the Canadian mine should probably be the correct one.
3.Record absolute inches under sea level a person has reached, including underwater immersion.
4.Record closest a human has come to Earth's center.
The difference between equatorial and polar radius is 22 km, so you might search for the answer at the Poles. I would not recommend the traveler to chose Antarctica, as it has an average elevation around 2000 meters. And I would say he would fails again at North Pole against Victor Vescoso's Marianas dives -he goes "on foot", he is not a submariner :)-. Perhaps a military submarine that is traveling around the Arctic Ocean may be the winner. You will need to make some calculations to determine if the Arctic Seabed is closer to Earth's Center than the Marianas Trench.
5.Record depth from the surface a person has reached "on foot".
If you consider miners that take an elevator to still be going "on foot", the South Africa ones should have that record. If not, a spelunker.
6.Record absolute depth under sea level a person has reached "on foot".
If you consider Vescoso to have been "on foot", he wins again. If not, and you consider miners going to their jobs as being "on foot", then it would be the Canadian miners (2.65 km below see level). If you are strict against both, my first thougth is then maybe port workers (note a submariner won't win in this case neither), or maybe a spelunker, but user Semidiurnal Simon clarifies it on comments. I was wrong (I said I made the estimations quickly) as: "For the strictest "on foot", it won't be port workers, it'll be somebody in a below-sea-level basin (e.g. by the Dead Sea), or possibly a low-altitude mine that has a drift (slanted corridor) entry and so doesn't require an elevator."
7.Record absolute depth under surface by drilling.
Sending machines from the surface (by borehole) rather than humans, the Kola Superdeep Borehole is the deepest (12km).
8.Record closest drill to Earth's Center.
The Ocean Drilling Program could have this record, but I cannot determine where. Average seabed deep rounds -4.000 m. and the Arctic Ocean is not a deep ocean in comparison with the Pacific and Atlantic. So the Kola Borehole may well have this record too.
"Deepest drillings. The Kola Superdeep Borehole on the Kola peninsula of Russia reached 12,262 metres (40,230 ft) and is the deepest penetration of the Earth's solid surface. The German Continental Deep Drilling Program at 9.1 kilometres (5.7 mi) has shown the earth crust to be mostly porous. Drillings as deep as 2.1 kilometres (1.3 mi) into the seafloor were achieved at DSDP/ODP/IODP Hole 504B. Because the continental crust is about 45 km thick on average, whereas oceanic crust is 6-7 km thick, deep drillings have penetrated only the upper 25-30% of both crusts.
*Distorted scale. You can take a look at this image to have an idea how thick the crust is at Earth scale.
Here is some information about a Japanese plan to reach the mantle.
There was a proposal by geophysician David J. Stevenson that was supposed to go to the Earth's Core, but it was controversial. It was published on Nature on 2003.
I encourage students to try to solve the question raised in option 4 to determine:
Which is closer to Earth's Center, the Marianas Trench or the North Pole's seabed?
The book "Blind Descent" by James M. Tabor details this answer and I found it a terrific read. I can't recall the exact depth, but what stuck in my mind was one particular explanation where the author said that we (humans) have gone as far below sea-level as above it. That is, roughly equivalent to the height of Everest - well over 8km.