JavaScript (Node.js), 68 bytes

f=([c,d,...s],a=b=0)=>c?f(s,a-14%('0x'+c)%7*d,b-=d):(a/b).toFixed(2)

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JavaScript (ES6),  72  70 bytes

Input format: A1B3C2F3B4

f=([c,d,...s],a=b=0)=>c?f(s,a-`0x${c+c}0`%7%6*d,b-=d):(a/b).toFixed(2)

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Grade conversion

 grade | hexa  | decimal | mod 7 | mod 6
-------+-------+---------+-------+-------
   A   | 0xAA0 |   2720  |   4   |   4
   B   | 0xBB0 |   2992  |   3   |   3
   C   | 0xCC0 |   3264  |   2   |   2
   D   | 0xDD0 |   3536  |   1   |   1
   F   | 0xFF0 |   4080  |   6   |   0

Perl 5, 57 53 50 + 1 (-n) = 59 55 51 bytes

/ /;$c+=$';$\+=$'*(69+/F/-ord)}{printf'%.2f',$\/$c

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Input format: line separated, credits followed by grade:

grade credits

Example:

A 3
B 4
A 1
A 1

Julia 0.6, 46 43 42 bytes

g%h=round(max.(69-Int.(g),0)⋅h/sum(h),2)

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Explanation

Input format: g: vector of grades; h: vector of credit hours

• g%h: Redefine % operator.
• 69-Int.(g): Convert 'F','D','C','B','A' to -1,1,2,3,4 respectively for each element of g.
• max.( ,0): Clamp range to 0:4 (element-wise).
• The rest is simple vector math.
• Rounding costs 9 bytes.

Java 1.8, 287 249 Bytes

-38 bytes thanks to Bumptious

Golfed

static String N(String[]j){float g=0;float b=0;for(int i=0;i<j.length;i+=2){g=((m(j[i])*Float.parseFloat(j[i+1])+g));b+=Double.parseDouble(j[i+1]);}return String.format("%.2f",g/b);}static float m(String l){return l.equals("F")?0:('E'-l.charAt(0));}

Ungolfed

interface C {
static void main(String[] z) throws Exception {
    String[] j = {"A", "4", "B", "3", "C", "2", "D", "1", "F", "1"};
    System.out.println(N(j));
}

static String N(String[] j) {
    float g = 0;
    float b = 0;
    for (int i = 0; i < j.length; i += 2) {
        g = ((m(j[i]) * Float.parseFloat(j[i + 1]) + g));
        b += Double.parseDouble(j[i + 1]);
    }
    return String.format("%.2f", g / b);
}

static float m(String l) {
    return l.equals("F") ? 0 : ('E' - l.charAt(0));
}
}

Java, 211 bytes

Input format: A1B3C2F3B4

Golfed

interface A{static void main(String[] a){int p=0,t=0,h=0,s=0;for(int c:a[0].toCharArray())if(p++%2==0)t=c=='A'?4:c=='B'?3:c=='C'?2:c=='D'?1:0;else{s+=(c-=48)*t;h+=c;}System.out.print(Math.ceil(100d*s/h)/100);}}

Unglofed

static void main(String[] a) {
    int p=0, //position in string
    t=0, //temp var, used to store the grade between iterations
    h=0, //credit sum
    s=0; //sum of weighted grade

    for(int c:a[0].toCharArray())
        if(p++%2==0)
            //map c to grade value, assign to temp variable t
            t=c=='A'?4:c=='B'?3:c=='C'?2:c=='D'?1:0;
        else{
            //map c to credit value, add temp variable (grade from previous char) * value of this char (credit) to sum
            s+=(c-=48)*t;
            //also, add credit to credit sum
            h+=c;
        }
    System.out.print(Math.ceil(100d*s/h)/100); //grade sum / credit hours sum, to 2dp*/
}

Other version

My gut frealing told me that using a different input format (ABCF1324) would make the code shorter. It seems like it didn't. The version below is 234 bytes long.

Golfed

interface A{static void main(String[] b){char[] a=b[0].toCharArray();int l=a.length/2,h=0,s=0,g,c,i;for(i=0;i<l;i++){g=a[i];g=g=='A'?4:g=='B'?3:g=='C'?2:g=='D'?1:0;c=a[i+l]-48;s+=g*c;h+=c;}System.out.print(Math.ceil(100d*s/h)/100);}}a

Ungolfed

static void main(String[] b) {
    char[] a=b[0].toCharArray(); //char array
    int l=a.length/2, //first grade char
    h=0, //credit sum
    s=0, //sum of weighted grade
    g,c, //avoid declaration in for loop. grade and credit being iterated
    i; //avoid declaration in for loop
    for(i=0;i<l;i++) {
        g=a[i];//get char representing grade from array
        g=g=='A'?4:g=='B'?3:g=='C'?2:g=='D'?1:0; //convert to grade
        c=a[i+l]-48;//get char representing grade from array and convert to credit (48 is value of '0')
        s+=g*c; //add weighted grade to sum
        h+=c; //add credit to sum
    }
    System.out.print(Math.ceil(100d*s/h)/100); //grade sum / credit hours sum, to 2dp*/
}

R, 64 bytes

function(G,H)sprintf("%.2f",(5-match(G,LETTERS[-5]))%*%H/sum(H))

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thanks to user2390246 for fixing a bug!

Jelly,  15  21 bytes (12 with no rounding)

+6 bytes for the strict formatting (almost certainly possible in less but it's bed time)

Oạ69.Ḟ×S×ȷ2÷⁹S¤RLDż”.

A full program taking the grades and the respective credit hours which prints the calculated GPA (Note: the rounding method is to floor, as allowed in the OP).

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With no rounding for 12 bytes:

Oạ69.Ḟ×S÷⁹S¤

How?

Oạ69.Ḟ×S×ȷ2÷⁹S¤RLDż”. - Link: list of characters, grades; list of number, creditHours
                      -                                   e.g. "AFBDC", [5, 2, 4, 1, 2]
O                     - cast to ordinals (vectorises)          [65,70,66,68,67]
  69.                 - literal 69.5
 ạ                    - absolute difference (vectorises)       [4.5,0.5,3.5,1.5,2.5]
     Ḟ                - floor (vectorises)                     [4,0,3,1,2]
      ×               - multiply by creditHours (vectorises)   [20,0,12,1,4]
       S              - sum                                    37
         ȷ2           - literal 100
        ×             - multiply                               3700
              ¤       - nilad followed by link(s) as a nilad:
            ⁹         -   chain's right argument, creditHours  [5, 2, 4, 1, 2]
             S        -   sum                                  14
           ÷          - division                               264.2857142857143
               R      - range                                  [1,2,3,...,264]
                L     - length                                 264
                 D    - digits                                 [2,6,4]
                   ”. - literal '.'
                  ż   - zip together                           [[2,'.'],6,4]
                      - implicit print (smashing)              2.64

Wolfram Language (Mathematica), 39 bytes

N[(5-LetterNumber@#2/.-1->0).#/Tr@#,3]&

Takes a list of credit hours, and then a string of grades.

Does not work on TIO because TIO uses Mathematica kernel (which doesn't want to print arbitrary precision numbers)

Python 3, 66 bytes

-5 bytes thanks to Rod.

lambda g,c:'%.2f'%sum('FDCBA'.find(i)*j/sum(c)for i,j in zip(g,c))

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Python 2, 69 bytes

lambda x:'%.2f'%sum('FDCBA'.find(a)*b*1./sum(zip(*x)[1])for a,b in x)

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