Use a for-loop

using System;

public static class Hamming
{
    public static int Distance(string strand1, string strand2)
    {
        if (strand1.Length != strand2.Length)
            throw new ArgumentException("Strands have different length");

        var distance = 0;

        for (var i = 0; i < strand1.Length; i++)
        {
            if (strand1[i] != strand2[i])
                distance++;
        }

        return distance;
    }
}

The first step is to check if both strings have the same length:

if (strand1.Length != strand2.Length)
    throw new ArgumentException("Strands have different length");

Once we've verified that, we define a variable that will keep track of the hamming distance between the two strings:

var distance = 0;

We then use a for-loop to iterate over each index of the first (and second) string:

for (var i = 0; i < strand1.Length; i++)

Within the loop, we can then use a simple if-statement to check if the two strings have different characters at the specified index, and if so, increment the distance by one:

if (strand1[i] != strand2[i])
    distance++;

Finally, once each index has been checked, we return the hamming distance:

return distance;