|MadSci Network: Virology|
The short answer: Viruses can be made of RNA or DNA but they rely upon the machinery of the cells they invade to make their protein coats.
It might be helpful to think of the genetic "equation" which goes: DNA => RNA => protein. Which is to say that our genes are made of DNA which is "transcribed" into RNA which then is "translated" into protein.
RNA viruses come in two types. Some RNA viruses can be directly transcribed into protein while others (such as HIV) are reverse transcribed into DNA and integrate into the chromosomal DNA. RNA is then made from the integrated DNA and then protein from the new RNA.
DNA viruses also may or may not require integration into chromosomal DNA. To make the protein coat, the viral DNA is transcribed into RNA and then translated into the protein.
A more "complete" explanation can be found in any virology books you can find in your local college library.
The viral particles are assembled within the infected host cell. Nucleic acid (either RNA or DNA) is then packaged with particular viral proteins produced within the cell, and in many cases, a covering lipid membrane. Once released, the viral particle then has everything it needs to infect more cells.
Try the links in the MadSci Library for more information on Virology.