It's not exactly slang. It is more like a phonetic spelling of how some people actually sound when they say "I'll have." The "I'll" part is not slang, but is a contraction of "I will." It is less formal than "I will," but is not slang. The "' 'ave " part looks like it is trying to phonetically spell the actually sound of a person saying "I"ll have." That's because some people, especially some English and Australian people but some Americans too, drop the "h" sound at the beginning of "have" when they are speaking.

I'd add, that here in American the apostrophe in front of the 'ave' in "I'll 'ave", indicates that a letter (s) has been dropped in the word. It has been spelled phonetically to reflect how the person is 'literally' speaking. I found this info which might be helpful, it explains the use of the apostrophe to indicate a 'glottal stop'... To stand for a glottal stop. This is an extension of the first one: If you drop a letter such as h or t when you're speaking, you may replace it with a glottal stop — just a momentary catch in the breath. Cockney speech is often represented this way: 'Ere! Wot's the ma''er? So when we're writing down words in another language that have a glottal stop, we typically use the apostrophe to represent it http://theweek.com/articles/583980/wacky-world-apostrophes-explained

Dropping the H is very common for have in British English. It is not really slang more a pronunciation issue. In the north, we drop the H more than we say it.

It is not "the slang"... remember that slang is not standard, and that means not everyone speaks in that way. Here's the quote in context: - "I'll 'ave that bag," she hissed. Luckily, I was wearing all my clothes to keep warm. Notice that " 'ave " is not part of " I'll ". Some dialects drop the H sound. Even in standard English, sometimes the H sound is weakened in unstressed syllables. This is simply how that particular woman spoke.

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