Modify & replace:
One option is to mask the class with a modified copy (modify code, recompile code, add modified classes to the classpath before patched classes), similar to the approach used here to inspect how a normally unavailable method works.
If you do not have sources to modify, you can "reverse" almost any .class file into more-or-less readable source code using decompilers. Notice that, depending on licensing, you may not have permission to do so and/or to redistribute your changes.
Patch via agent:
You can also patch the methods using the -javaagent:<jarpath>[=<options>]
commant-line option. The "agent" is a jar that gets to modify loaded classes and alter their behaviour. More information here.
Mock:
If you have control over where the methods are called, you can replace the target instance with a stubbed version. Libraries such as Mockito make this very, very easy:
LinkedList mockedList = mock(LinkedList.class);
// stubbing appears before the actual execution
when(mockedList.get(0)).thenReturn("first");
Even though Mockito does not support mocking private methods natively (mostly because it is considered bad manners to look at other classes' privates), using PowerMock allows you to do so (thanks, @talex).