Source a recording of the public speech you want to use in your song. Ensure that the speech recording that you sample is in the public domain; otherwise, you must ask the owner of the recording for permission to use the speech. All of the speeches that federal government employees -- including Presidents -- deliver in an official capacity are in the public domain. The Miller Center at the University of Virginia hosts an extensive set of audio recordings of Presidential speeches, while the National Archives has collected a vast sound recording archive, much of which is in the public domain.
Edit the speech down to a manageable sample. Listen to the speech closely, then select the most memorable and impactful section to use in your song. Most public speeches -- especially political speeches -- contain a call to action: "Ask not what your country can do for you, but what you can do for your country," for example. This call to action is usually the best part of a speech to use in a song.
Position the sample of the speech within the framework of your song. The exact way in which you use the speech will, of course, depend on the genre of music that you're making; in general, however, speech samples work well as transitional points in a song. If you're making a dubstep track, for example, use the public speech sample to introduce the bass drop; in a trance song, introduce the speech sample near the end of the breakdown, leading into the point at which the drums re-enter the song.
Mix the public speech sample with the rest of your song. The audio fidelity of many speech recordings, especially older ones, is low; you may need to apply heavy equalization and noise reduction to make the sample fit into a modern-sounding song. If you're mixing the speech sample with vocals that you've recorded yourself, apply the same compression and reverb to all of the vocal tracks in the song. This evens out the volume dynamics and ensures that the speech sample doesn't sound disjointed.