How to implement this FilteringIterator?

If you want to do it yourself, you can use code similar to what I've written below. However, I do recommend you use Guava's Iterators.filter(Iterator, Predicate)

public class FilteredIterator<T> implements Iterator<T> {
    private Iterator<? extends T> iterator;
    private Filter<T> filter;
    private T nextElement;
    private boolean hasNext;

    /**
     * Creates a new FilteredIterator using wrapping the iterator and returning only elements matching the filter.
     * 
     * @param iterator
     *            the iterator to wrap
     * @param filter
     *            elements must match this filter to be returned
     */
    public FilteredIterator(Iterator<? extends T> iterator, Filter<T> filter) {
        this.iterator = iterator;
        this.filter = filter;

        nextMatch();
    }

    @Override
    public boolean hasNext() {
        return hasNext;
    }

    @Override
    public T next() {
        if (!hasNext) {
            throw new NoSuchElementException();
        }

        return nextMatch();
    }

    private T nextMatch() {
        T oldMatch = nextElement;

        while (iterator.hasNext()) {
            T o = iterator.next();

            if (filter.matches(o)) {
                hasNext = true;
                nextElement = o;

                return oldMatch;
            }
        }

        hasNext = false;

        return oldMatch;
    }

    @Override
    public void remove() {
        throw new UnsupportedOperationException();
    }
}

public interface Filter<T> {

    /**
     * Determines whether elements should be filtered or not.
     * 
     * @param element the element to be matched against the filter
     * @return {@code true} if the element matches the filter, otherwise {@code false}
     */
    public boolean matches(T element);
}

If this is homework, this won't help you, but if not: the Guava Library has the exact functionality you are after

Iterators.filter(Iterator, Predicate)

(You might have a look at how they did it for inspiration)

You would need to make your iterator stateful. Cache the last value you retrieved from hasNext and use that from the next method, if it exists.

private boolean hasCached;
private T cached;

public boolean hasNext() {
   if ( hasCached ) return true;
   //iterate until you find one and set hasCached and cached
}

public T next() {
   if ( hasCached ) {
      hasCached = false;
      return cached;
   }
   //iterate until next matches
}