Hell0 W0rld! scored by ASCII values

Lenguage, score 0

Well, this is quite boring and obvious, but it seems to achieve the perfect score (assuming I understand the challenge correctly). To fix this loophole, you should probably add the program's length to the sum of the code points; I'm pretty sure that won't break the relative order of the current answers.

The program is

2282524454641945458940768311481012240407465625061984267694778013056442426959468031225640633070772676474431950201473533908

null bytes. Fortunately, 0 multiplied by the number above is still 0.

I believe this decompresses (alternatively, it compresses) to the automatically generated (with the help of a random web page that claims "Original Java program not from me") brainfuck program:

-[------->+<]>-.-[->+++++<]>++.+++++++..[-->+<]>------.[--->++<]>.---[->+++<]>.+[-->+<]>++++.[----->+<]>++.------.--------.-[--->+<]>.

It almost certainly could be golfed (but I haven't tried), but that is unnecessary.

05AB1E, score 513 252

^{-261 points thanks to @Dorian}

Posting this separately from my other answer, as this one is arguably less cheating. I've been waiting for somebody else to do this for a while, but nobody did. I tried multiple languages. Pyth has extremely short code for this (Cl"...", if I remember correctly), but it turned out it can't input null bytes, and most other golfing languages I found have painfully large char codes for quotes (and I couldn't write this piece of code in the rest).

"ǝǝǝǝǝǝǝǝǝǝǝǝ...ǝǝǝǝǝ"gтB

Where there are 174047470094320053473995 ǝ characters, that conveniently happen to have a value of 0 in the code page.

Explanation:

"ǝǝǝǝǝǝǝǝǝǝǝǝ...ǝǝǝǝǝ"        push the long string
                      g       pop and get its length
                       тB     push 100 and convert to that base

You definitely shouldn't try the full version online, but you can try online the version with the length finding part skipped.

7, 41 characters, code point sum 81

32002453003001200522231203103002440537403

Try it online!

Unlike with many of my 7 programs, there's nothing particularly clever going on here; this basically just prints a string in the simplest possible way (which in 7 involves encoding the string into a piece of source code, evaluating it, analysing the resulting stack element to figure out what source code is most likely to have created it, and then giving the resulting source code directly to the output routines).

The reason this scores so well is that 7 has a code page consisting of only 8 characters, and the code that's used to represent string literals tends to use only 0, 1, 2, 3, 4, 5 (as those commands have much easier-to-analyse effects on the stack than 6 and 7 do, so it's more likely that the original source will be accurately reproduced by the 0 command). This means that none of the code points in the bulk of the program will go above 5 (the assignment of code points to characters in 7 is the obvious one), leading to an average code point value of around 2½ (for this program, happenstance means it's more like 2). Thus, although there are a lot of characters in the program, the sum of the codepoints continues to stay low. (Unlike the Lenguage answer, the code point sum + character count is also very low, so this answer isn't even really cheating.)