The easiest way to do this, because you're subtracting a bigger number from a smaller number is to treat it as if the smaller number was being subtracted from the larger, then, when you've worked it out, you put a minus in front of the result as the very last thing you do.
So let's do that. We work out 5 3/5 - 3 1/3.
Subtract the whole numbers first: 5-3=2 and make a note of the 2.
The fractions are the next problem, because they have different denominators. They need the same denominator to make subtraction easy. What number do 3 and 5 both go into? The smallest is best. They both go into 15. Take the fraction 3/5. 15 divided by 5 is 3, multiplied by the numerator is 9, so 3/5 is 9/15. Now take 1/3. 15/3 is 5 multiplied by the numerator 1 is still 5, so 1/3 is the same as 5/15. Now both fractions have the same denominator, 15. We have 9/15 take away 5/15=4/15, just as 9 apples take away 5 apples leaves 4 apples. We have fifteenths instead of apples but the same rule applies. Now we can add in the whole number 2 we got earlier so we have 2 4/15. Finally, put in the minus: -2 4/15 is the answer.