Yes, Surely you can pursue BCA from private colleges. You can enroll in any of private colleges by your class 10(+2) merits or by giving Entrance Examinations like KIITEE BCA,LUCSAT BCA PESSAT .There are many good private collegesto pursue BCA.

For choose college and to get more information regarding colleges you can refer to this link :

https://it.careers360.com/colleges/list-of-bca-colleges-in-india