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Maximizing performance

A LINQ variation

This can be further shortened using LINQ:

var sum = Enumerable.Range(0,1000)
              .Where(i => i % 3 == 0 || i % 5 == 0)
              .Sum();

LINQ is just a nicer syntaxt, but it uses a for/foreach iteration in the background, so I suspect it won't be much more performant than the previous example. But I do consider this highly readable.

Maximizing performance

You will need to iterate over the multiple of 5 separately. However, should you keep using this approach, I would always suggest splitting these loops anyway.

// Check every number from 0 to 1000 (not including 1000)
for(int i = 0; i < 1000; i++)
{
    var isMultipleOf3 = i % 3 == 0;
    var isMultipleOf5 = i % 5 == 0;

    //If it is divisible by 3 or it is divisible by 5
    if(isMultipleOf3 || isMultipleOf5)
    {
        //then add it to the sum
        sum += i;
    }
}

You don't need to use the booleans. I simply added them to simplify the example. if(i % 3 == 0 || i % 5 == 0) would be equally okay to use because it's still reasonably readable.
If the calculations become more complex, I suggest always using the booleans so you neatly break your algorithm down to small and manageable steps. It will do wonders for your code readability.

// Check every number from 0 to 1000 (not including 1000)
for(int i = 0; i < 1000; i++)
{
    var isDivisibleBy3 = i % 3 == 0;
    var isDivisibleBy5 = i % 5 == 0;

    //If it is divisible by 3 or it is divisible by 5
    if(isDivisibleBy3 || isDivisibleBy5)
    {
        //then add it to the sum
        sum += i;
    }
}

You don't need to use the booleans. I simply added them to simplify the example. if(i % 3 == 0 || i % 5 == 0) would be equally okay to use because it's still reasonably readable.

If the calculations become more complex, I suggest always using the booleans so you neatly break your algorithm down to small and manageable steps. It will do wonders for your code readability, and it does not impact performance (the compiler will optimize this in a release build).