Because only you, as an actor, know what you are capable of, it is up to you to determine how best to play the character. With that being said there are some suggestions based on the lines themselves that could help you understand the character of Butch Weldy. First and foremost your assumption of emotionless is close to where you want to be in your performance, however, based on the this poem being a recounting of his blinding and denial of compensation Butch should have a sort of earnest and resigned tone.
Weldy’s lines begin with, “After I got religion and steadied down”. This line gives the reader insight into Butch, suggesting that he prior to this point was a bit uncivilized in his actions. Knowing Butch’s earlier status could suggest that he should be played with a hint of underlying rage. Think of a man who is trying to keep himself in check and not revert to his previous self. This keeping himself in check adds to the resigned tone. He knows that no matter what has happened to him he can’t become the man he was.
The next few lines are very straight forward and matter of fact.
“They gave me a job in the canning works,/ And every morning I had to fill/ The tank in the yard with gasoline,/ That fed the blow-fires in the sheds/ To heat the soldering irons./ And I mounted a rickety ladder to do it,/ Carrying buckets full of the stuff./ One morning, as I stood there pouring,/ The air grew still and seemed to heave,/ And I shot up as the tank exploded,/ And down I came with both legs broken,”
This shows the reader that Weldy is earnest—he does not embellish what has happened to him. He does not try to make the listeners of his story feel pity for him. It is not difficult to imagine Butch Weldy speaking in a slow matter of fact way.
What follows is where in your reading you can start to adjust your tone.
“And my eyes burned crisp as a couple of eggs.”
Here is where you could introduce a hint of sorrow; this line in its use of metaphor represents figurative language that can be more emotional.
Butch catches himself in the next few lines and goes back to the more matter of fact reporting.
“For someone left a blow-fire going,/ And something sucked the flame in the tank.”
Recounting the judge’s statement is where you could introduce some disbelief, since this is Butch recounting the judge’s decision the disbelief will have had time to sink in and would be read more like resignation than shock.
"The Circuit Judge said whoever did it/ Was a fellow-servant of mine, and so/ Old Rhodes’ son didn’t have to pay me./ Lastly, you may want to shift slightly from resignation to sorrow for the reading of the last few lines./ And I sat on the witness stand as blind/ As Jack the Fiddler, saying over and over,/ 'I didn’t know him at all.'”