"An awful lot": Any negative connotation?

Today was my first time I saw people use "an awful lot of" instead of "a lot of".

It reminded me of "terribly good", which obviously has little negative connotation. But how about "an awful lot"?

Does the person saying it imply that he hates the state of there being too many of something?

It depends on what is being described. An awful lot of trouble is negative. An awful lot of good is positive.

I've only heard "that did an awful lot of good" as sarcastic - it didn't do much good at all. If you're in "an awful lot of trouble," you are in trouble.

The phrase an awful lot of is neutral when it uses awful in the sense "Exceedingly great; usually applied intensively. [E.g.] an awful bonnet [and] I have learnt an awful amount today." This sense applies if the phrase is not used sardonically. For example, sardonic tone could cause "What an awful lot of fish!" to mean "What a bad bunch of fish!" instead of "What a vast quantity of fish!"

As noted in etymonline,

c.1300, agheful "worthy of respect or fear," from aghe, an earlier form of awe, + -ful. Replaced O.E. egefull. Weakened sense "very bad" is from 1809; weakened sense of "exceedingly" is by 1818.

Thus, the idea of "awe-inspiring" lent itself to the sense noted above of awful.

Regarding lot of: one of my high-school English teachers marked down sentences containing lot of, but that may have been personal preference on her part. I don't know of any specific grammar rules against it.

There's a Frank Sinatra's song named: "The Coffee Song"

He sings about the amount of coffee Brazil used to produce and sell in the 50's and 60's (and it still does). Brazil actually produces one of the best coffees in the world, but it only keeps the worst for itself. Alright, enough of history!

He sings in the chorus: "They've got an awful lot of coffee in Brazil"

In that sense, he means: "They've got A LOT of coffee in Brazil".

So it's clear that he doesn't mean any negative thing!

Cheers!