That’s just eye-dialect for boil. The character Joe is speaking with a non-standard or rustic diction and accent, and Dickens is using funny spellings to show that.
In the broader passage surrounding that particular quotation I give below, I’ve set such deviations from Standard English in bold:
“Though mind you, Pip,” said Joe, with a judicial touch or two of the poker on the top bar, “rendering unto all their doo, and maintaining equal justice betwixt man and man, my father were that good in his hart, don’t you see?”
I didn’t see, but I didn’t say so.
“Well!” Joe pursued, “somebody must keep the pot a biling, Pip, or the pot won’t bile, don’t you know.”
I saw that, and said so.
“ ’Consequence, my father didn’t make objections to my going to work; so I went to work at my present calling, which were his too, if he would have followed it, and I worked tolerable hard, I assure you, Pip. In time I were able to keep him, and I kep him till he went off in a purple leptic fit. And it were my intentions to have had put upon his tombstone that Whatsume’er the failings on his part, Remember reader he were that good in his hart.”