Computing the circumference of two circles
Although your code works, it's not so good.

Declare variables as late as possible

Here you declare some variables at the top, and use them much later, for example smallCircum:


double smallCircum;
double largeCircum;
InputHandler input = new InputHandler();
Question5 myq = new Question5();

double a = input.getDouble("Radius for the smaller circle:");

smallCircum = (myq.start(a));//calculate the small circle circumference



It would be better to declare as late as possible, right before you need it:

InputHandler input = new InputHandler();
Question5 myq = new Question5();

double a = input.getDouble("Radius for the smaller circle:");
double smallCircum = (myq.start(a));//calculate the small circle circumference


Pointless field

In Question5, the variable circumference doesn't need to be a field.
It's only used here:


private double circum(double radius){
    circumference = 2 * Math.PI * radius;
    return circumference;
}



But there is no need to store that value in a field, and no need for even a local variable, you can simply return it directly:

private double circum(double radius) {
    return 2 * Math.PI * radius;
}


static method

It's unclear why this method in InputHandler is static:


public static double getDouble(String prompt){



I suggest to drop the static keyword, make it non-static.

Poor naming

Looking at this:


smallCircum = (myq.start(a));//calculate the small circle circumference



A method named start returns the calculated circumference.
That's really not intuitive.
Why not rename start to calculateCircum?
If its containing Question5 class needs to be kept general purpose and not imply circumference, then perhaps you can rename the method to just calculate instead.
Probably almost anything is better than start.

Another thing, a is a poor name to store radius.

Make private what you can

In Question5, df should be private.

Make final what you can

In Question5, df can be final.

Suggested implementation

With the above suggestions applied, the solution becomes:

package com.janosgyerik.practice.oj.leetcode.hard.SubstringWithConcatenationOfAllWords;

class Q5Main {
    public static void main(String args[]) {
        InputHandler input = new InputHandler();
        Question5 myq = new Question5();

        double smallRadius = input.getDouble("Radius for the smaller circle:");
        double smallCircum = myq.calculate(smallRadius);

        double largeRadius = input.getDouble("Radius for the larger circle: ");
        double largeCircum = myq.calculate(largeRadius);

        myq.showDetails(smallCircum, largeCircum);
    }
}    

class Question5 {
    private final DecimalFormat df = new DecimalFormat("0.000");

    public double calculate(double value) {
        return circum(value);
    }

    private double circum(double radius) {
        return 2 * Math.PI * radius;
    }

    public void showDetails(double smallCircum, double largeCircum) {
        System.out.println("Circumference of smaller circle: " + df.format(smallCircum));
        System.out.println("\n");
        System.out.println("Circumference of larger circle: " + df.format(largeCircum));
        System.out.println("\n");
        System.out.println("Difference: " + df.format((largeCircum - smallCircum)));
    }
}    

class InputHandler {
    private static final String DOUBLE_DEFAULT_PROMPT = "Enter integer: ";

    public double getDouble() {
        return getDouble(DOUBLE_DEFAULT_PROMPT);
    }

    public double getDouble(String prompt) {
        String inStr = JOptionPane.showInputDialog(null, prompt);
        return Double.parseDouble(inStr);
    }
}