Simple:

 @Test 
 public void whenSerialNumberIsEmpty_shouldThrowSerialNumberInvalid() throws Exception {
  try{
     whenRunningSomething_shouldThrowMyExceptionWithInternalErrorCode();     
     fail("should have thrown");
  }
  catch (MyExceptionClass e){
     assertThat(e.getCode(), is(MyExceptionClass.INTERNAL_ERROR_CODE));
  }

That is all you need here:

• you don't want to expect that specific exception, as you want to check some properties of it
• you know that you want to enter that specific catch block; thus you simply fail when the call doesn't throw
• you don't need any other checking - when the method throws any other exception, JUnit will report that as error anyway

You can check for it using hamcres matchers as long as thrown.expect is overload to receive Matcher

thrown.expect(CombinableMatcher.both(
           CoreMatchers.is(CoreMatchers.instanceOf(MyExceptionClass.class)))
           .and(Matchers.hasProperty("errorCode", CoreMatchers.is(123))));

Note that you will need to add hamcrest matcher to your dependencies. Core matched that are included in JUnit is not enough.

Or if you don't want to use CombinableMatcher:

thrown.expect(CoreMatchers.instanceOf(MyExceptionClass.class));
thrown.expect(Matchers.hasProperty("errorCode", CoreMatchers.is(123));

Also, you don't need (expected = MyExceptionClass.class) declaration for @Test annotation

Expanding upon Sergii's answer, you can clean this up even more by writing a custom matcher.

import org.hamcrest.Description;
import org.hamcrest.TypeSafeMatcher;

public class CustomMatcher extends TypeSafeMatcher<CustomException> {

    public static CustomMatcher hasCode(String item) {
        return new CustomMatcher(item);
    }

    private String foundErrorCode;
    private final String expectedErrorCode;

    private CustomMatcher(String expectedErrorCode) {
        this.expectedErrorCode = expectedErrorCode;
    }

    @Override
    protected boolean matchesSafely(final CustomException exception) {
        foundErrorCode = exception.getErrorCode();
        return foundErrorCode.equalsIgnoreCase(expectedErrorCode);
    }

    @Override
    public void describeTo(Description description) {
        description.appendValue(foundErrorCode)
                .appendText(" was not found instead of ")
                .appendValue(expectedErrorCode);
    }
}

The error code can then be checked like:

import org.junit.rules.ExpectedException;

public class MyObjTest {

    @Rule
    public ExpectedException thrown = ExpectedException.none();

    @Test
    public void someMethodThatThrowsCustomException() {
        thrown.expect(CustomException.class);
        thrown.expect(CustomMatcher.hasCode("110501"));

        MyObj obj = new MyObj();
        obj.methodThatThrowsCustomException();
    }
}

Reference: https://dzone.com/articles/testing-custom-exceptions