Find the tempo, or beats per minute, of the original song you are attempting to remix. This will allow you to create rhythms that match up accordingly when laying parts of the original material over them. If you have received the original source tracks, the individual instrument recordings for the song from the artist with the intent of remixing, the tempo should be provided in an attached document.
If you cannot find the original tempo, you may have to make adjustments. Recording programs such as Sony Acid Pro and Pro Tools can "beatmap" the master tracks, which will determine an estimated tempo and adjust the recording to it. If this isn't an option for you, you can still create beats, but you will have to use your ears to stretch them out to match the desired speed.
Create the drumbeats using a hardware drum machine. External drum stations provide a variety of drum sounds, can be used to make and sequence loops, and can also be connected to a computer via a MIDI port to control digital drum machines. Boss, Roland and Akai as well as other companies sell quality drum machines for between $200 and $600. They can also be purchased used at a significantly lower cost.
Use a software synthesizer program not only to create drumbeats, but also to implement other sound effects or sample beats that were recorded live. This is a great option if you are sampling drums from the original recording. Programs like Propellerhead Reason and FL Studio will read WAV files, a common format for uncompressed audio, and will let you use the files as samples in their numerous synthesizer and drum machine applications. They can also be synced to live recording software in something called "ReWire Mode." So if you want to create an entire production with them and then add live guitars, vocals or other instruments to your remix, you can do that as well.