Learn English – You’re not much use to me. Meaning

Here, "Okay doesn't even begin to cover it" means "The word okay is so inadequate that it does not even have the slightest chance of describing [topic]."

From Oxford Dictionaries Online:

begin

2. [no object with negative] informal Not have any chance or likelihood of doing a specified thing

cover

2. Deal with (a subject) by describing or analyzing its most important aspects or events

More generally, "doesn't even begin to..." is used when you want say that something is extremely lacking; that it does not have any chance of accomplishing the goal that it intends to. Some examples of proper ways to use "doesn't even begin to...", from the above source:

Indeed, even today, one cannot begin to comprehend the appalling fate suffered by these two young girls.

He concluded by saying that we couldn't even begin to comprehend what one day in his shoes would be like.

Clearly this too is an honour that most of you have little chance of even beginning to aspire to.

I've never heard this before, either, but it reminds me of the idiomatic expression six ways to Sunday, which, according to Wiktionary, means:

in every way imaginable

When discussing this phrase, WiseGEEK mentions:

Before becoming the title of a late 1990s mob-related movie staring Deborah Harry, the idiom "six ways to Sunday" was a less commonly used English expression meaning every way possible, thoroughly, or completely. Tracing the roots of such idioms is not always easy, [but] the meaning behind six ways to (or from) Sunday can be deciphered based on logical assumptions.

In terms of the calendar, there are six days after Sunday, or six days before Sunday, depending on the perspective. The phrase points out the inevitability of reaching Sunday, no matter what day serves as starting point.

Given that, I'd interpret the remark to mean:

You're dead – in every way imaginable!

meaning that death will inevitable for that character (unless something is done about the situation).