What is the origin of the phrase "[do something] like Hell"?

In the prepositional phrase "like hell", hell is an intensifier which is used to suggest a manner that uses all of the strength, speed, or effort that a person can summon. Its usage is from the mid/late 19 century.

• Pragmatically, "the hell" functions as an intensifier, usually indicating at least surprise, and often unpleasant surprise, and there is almost always another word in the sentence that receives contrastive stress to indicate that it is the focus of the surprise. Co-occurence of other interjections and topic particles like "well" is also common. The idea seems to be to mark the sentence strongly, iconic with the strength of the emotion to be expressed.

(www-personal.umich.edu/~jlawler)

Like hell:

• Recklessly, extremely, as in We ran like hell to catch the train. (Mid-1800s)
• Not at all, on the contrary, as in You think I'll call her stupid? Like hell I will! or Like hell I can't say that to Bob. (Late 1800s)

(The American Heritage® Dictionary of Idioms by Christine Ammer.)

Like hell:

• In an extravagant way; very forcefully : walleyes feed like crazy and ''bite like hell''/ Started screaming like hell (entry form 1855+, variant 1845+)

Note that:

• Variations: fun or shit may replace hell) Never; it is impermissible; in a pig's ass: Like hell you will!/ when the prostitute says ''Like fun you are'' (entry form 1925+, fun 1909+).

(The Dictionary of American Slang, Fourth Edition by Barbara Ann Kipfer, PhD. and Robert L. Chapman, Ph.D.)

In general terms, Like hell can be taken as very hard.