There are some odd checks in isvaliddouble() - you might want to think these through with a few example valid doubles. Why is it valid if it's one character long? Why do you need to check if it's one character long more than once? Why is the string valid if you come across a non-digit number? And, critically, what about the decimal separator?
Why are i and start double instead of int?
I'm not a fan of using exception handling instead of "else". However I can see that this could arguably be an exception so if you've been told to use exception handling then leave it in. Otherwise just use else.
Why are (complete entries) "+" and "-" valid? What would the value of these be? Zero?
You call atof() anyway, completely ignoring the return value of isvaliddouble(). Probably if isvaliddouble() returns FALSE you would want to skip the conversion and do something other than return a double - maybe return an error code indicating whether or not the conversion happened and pass in a pointer to a double, which would be used if the validity check passes.
Have you been told to write code to validate the double? atof() implements error checking so if you haven't been explicitly instructed to write code to perform double validation, you could just use the error checking built into atof().
You only check for a sign once but what about scientific notation? +1.2345e+5 is a valid number (and atof converts it correctly to 123450).