The central doctrine the Christian religion is belief that Jesus Christ was crucified, dead and buried, then rose from the dead. This is the Passion, remembered during Lent and Holy Week each year, and the Resurrection, celebrated on Easter Sunday.
The "orthodox" (official, accepted) belief is that Jesus rose from the grave physically, never to die again, so the answer to your question would be "yes, Jesus is still alive."
The Church teaches that Jesus rose in the same body that was crucified and died, that he remained in the grave from Friday evening through Sunday morning, when that same body rose as a "glorified" body that could never be harmed, injured, sickened or die again. The Church also teaches that Christ ascended bodily on Pentecost (50 days after Easter) and that he rules (bodily) in heaven until some unknown future day when he will (bodily) return to earth at the end of time.
However, since the earliest days of Christianity, there have been groups of believers who accept Jesus as Lord and Savior and follow his teachings without feeling that it is necessary to believe in the bodily resurrection. Many church members consider themselves Christians in every way, while openly or quietly believing that a miraculous resurrection simply isn't necessary.
From that viewpoint, the answer to your question would be, "yes, Jesus is alive in spirit, but not in a living physical body other than through those who believe in him."