In which part of the world do people use sentences like "I be doing this" (missing out the 'will' after the 'I')? Sounds like some of the 'street-ghetto' to me. What is it exactly?
It is not an accent, but a feature of some nonstandard dialects. As Peter Trudgill has written:
Standard English has irregular forms of the verb to be both in the present tense (am, is, are) and in the past (was, were). Many nonstandard dialects have the same form for all persons, such as I be, you be, he be, we be, they be, and I were, you were, he were, we were, they were.
In the United Kingdom, the use of be throughout the present tense is associated with the West Country.
This is an example of African American Vernacular dialect, not an accent. It expresses habitual behavior, as in "I be doing this every day" = "I do this every day" in standard American English.
Sounds like south-western Brit. Eng. to me. This explains why it reminded one person of pirate's speech, pirates generally being given the accent of Long John Silver, who, I think, was from Devon or Cornwall, although it is not clear from Treasure Island.