This is my new favorite question, because now I get to compare two of my favorite explosions side-by-side.
Let's assume we're limited to actual nuclear weapons that have been developed, and that we're using the one that will deliver absolutely the most powerful blow to the core of the Earth.
The Tsar Bomba is the most powerful and funnest-to-say nuclear bomb ever constructed by humans. When detonated, it delivered 50 Megatons of force, with a blast radius of 4 KM, and 240 Petojouls of energy. But that was only half of the maximum yeild for the bomb, limited to allow the payload delivery plane to escape the blast and for a scout plane to take pictures. If we take those safeties off, we can double all those figures.
...Which is still only one-half of what mother nature can do. In 1883, the volcanic island of Krakatoa erupted, blowing the entire island into pieces. This explosion had a force of 200 Megatons, four times that of the Tsar Bomba's explosion, and still twice the blast of a theoretical unhinged Tsar (but enough about Russian politics...). Both of these explosions occured essentially at the surface of the Earth, which helped the impact spread throughout the surface considerably.
If we could deliver a Krakatoa bomb to the surface of the Earth's core, what would happen?
Well, this Physics question (which is a good question, but strangely posted on Skeptics and therefore closed) points out that this type of explosion would be equal to an 8.75 earthquake. Which sounds frightening, unless you factor in the fact that earthquakes are (believed to be) the result of the constant churning of the mantle beneath the surface of the earth, and that the forces further down will be even stronger than that.
So unless your Alien Fusion Bomb manages to be stronger than the Krakatoa explosion, we aren't likely to feel an earth-shattering effect from just one. At worst, we may suffer from a few seismic disturbances, but you'll need to step up your payload considerably to do any real damage.