This would be done using a space crane.
A space crane is a type of sattilite placed in a geosynchronous orbit. It lowers cables and other equipment into the atmosphere.
While you could potentially mine diamonds this way, of equal or greater value would be the vast amounts of methane.
You would suck this up into space and place it in collosal space balloons (you could use the extreme cold of space to aid in liquifying it). You would transport the methane back to earth for use as fuel.
The balloons would be set in motion and basically float unmanned towards the earth, then captured and placed into Earth's orbit.
Your challenge would be getting the mining equipment in orbit around Neptune and getting it assembled. It would most likely need to be manned by a skeleton crew. Keeping them alive some 2.5 billion miles from home would be a logistical challenge.
Your next challenge would be controlling the equipment extending from space into Neptune's atmosphere.
To get such operations started in this day and age would require the efforts of several multi billion dollar corporations, and the government, and probably a 40 year turnaround for planning, engineering, R&D, production, transportation out, mining, then return trip of first balloon. Transportation alone is about a 20 year round trip.