If you scan the string left to right then the algorithm is fairly simple: sum=10*sum+(digit-'0'), so multiply what you've got by 10, then add the numeric value of the digit, which can be obtained by subtracting the ASCII code of '0' from the ASCII code of the digit in question, because in ASCII the digits are helpfully arranged so that '1'='0'+1, '2'='0'+2 and so on. And of course '0'-'0'=0. sum starts out as zero, of course.

So let's take the string '123'. sum=0, and the first character is '1'. '1'-'0'=1, so sum*10+1=1.
Second character is '2', so sum=1*10+2=12.
And so on to '3' where sum=12*10+3=123.

If you haven't got a multiply instruction then you can do shifts and adds to get the desired result. x*10=x*8+x*2, so you can shift x left 1, copy that to the result, then shift it left twice more and add that to the result, i.e.
shift left x 1 (x now contains 2* the original x)
add x to sum (sum now contains x*2)
shift left x 1
shift left x 1 (x now contains 8* the original x)
add x to sum (sum now contains x*2+x*8, or x*(2+8), or 10x).

If you want to be able to handle negative numbers as well then simply note somewhere if the first character is a minus, then at the end of the conversion just 2's complement the result (or flip all the bits and add 1, which is the same thing). So 123 is 7b in hex and 0111 1011 in binary, and -123 is 1000 0101.