If this is a question for a class, I do suggest you seek some clarification of the question. A situation in which someone would sue Lance Armstrong for the use of illicit drugs seems quite unlikely, and no one can sue another person for perjury. As you have stated this, these are both crimes. So, the question would appear to be whether Armstrong could be prosecuted in state or federal court. And the answer to that is that it depends. If Lance is in a particular state and in the United States and has violated the laws of either or both, they may both have jurisdiction, the state only may have jurisdiction, or the federal government only may have jurisdiction. Subject matter jurisdiction lies in a criminal court and is almost irrelevant to the inquiry, since people are prosecuted only in criminal court, not in civil court.
Most criminal law is the domain of the states, but there are federal laws concerning drug use, so if there is a federal law that preempts a state law, jurisdiction would lie in a federal criminal proceeding. For example, many states now allow the use of marijuana, either medically, recreationally, or both. But federal law makes the use of marijuana a crime. So even a person in a state that allows its use can be tried in federal court in a criminal prosecution. We say that the federal law preempts the state law in a situation like this. I do not know which performance enhancing drugs Armstrong has taken and which are actually illegal. For instance, it may or may not be illegal to do an autologous blood transfusion, one performance-enhancing means athletes use, in a particular state or in the United States crimes code.
Which court system would have jurisdiction regarding perjury is a function of to whom Armstrong perjured himself. If he lied to a state official under oath, that state would have jurisdiction over him for this crime. If he lied to a Congressman or a federal marshal, the federal government would have jurisdiction over him.
You can see how little subject matter jurisdiction matters under circumstances such as these. Different courts are allowed to try different kinds of cases, so that a bankruptcy would need to be filed with the federal bankruptcy court and if I sue another person over a private contract, the federal courts are not likely to have jurisdiction. If I seek an injunction, I must file in a court of equity. But if a person is being prosecuted for a crime, he or she must, of course, be prosecuted in criminal court.