Learn English – the way/as/how/like he did it. What’s the difference

Short Answer:

Aside from the third one (with how), all your sentences are correct and mean the same thing. The fourth (with like) is informal, and not normally used in serious speech or writing.

Long Answer:

All your sentences consist of two clauses, a main clause (I'll do it), plus a subordinate clause (he did it), and your question is about the terms that can join these two clauses, i.e., about subordinate conjunctions.

Let's take them one by one. The third is the only tricky one, so I'll save it for last.

This is the first definition OED provides for way:

a method, style or manner of doing something:
- way to do something That's not the right way to hold a pair of scissors.
- (informal, disapproving) That's no way to speak to your mother!
- way of doing something I'm not happy with this way of working.
- way (that…) It's not what you say, it's the way that you say it.
- I hate the way she always criticizes me.
...

As you can see, your first sentence perfectly fits this definition and its examples:

I'll do it the way (that) he did it.

This is a valid usage of way, among other constructions that you can use after this word. I don't think anybody's called way a conjunction, but in your sentence it acts like one, because of the hidden that which I hinted at in the parentheses.

As, simply, is a conjunction,

used to indicate by comparison the way that something happens or is done:
Dress as you would if you were having guests.
They can do as they wish.

Look at its definition: "...the way that something happens ..." . It suggests that there should be no difference between choosing as or the way in your sentence.

Like can be used as a conjunction too, but some people consider this use to be incorrect. We don't have to go that far, because it's so widespread in the informal register that you can't say it's wrong. It's just informal:

(informal)
In the same way that; as:
People who change countries like they change clothes.

Notice the as in the definition. So up to now we have three equivalent terms to use as a conjunction. (Watch out for the informality of like (instead of as) though!)

How is the tricky one here. It's certainly used as a conjunction sometimes, as in

You should remember how they fought.

But is it used for comparison? as in

*I'll do it how he did it.

Well, Michael Swan in his book Practical English Usage says no, we can't use how this way:

comparisons: how not used
In comparisons we use as or like or the way, not how.
Hold it in both hands as/like/the way Mummy does. (NOT ... how Mummy does)

So, aside from the third one (with how), all your sentences are OK, without any significant difference in meaning.
You can take a look at this Ngram diagram too:

See the hits for "do it how he". Almost none is relevant.