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.