I want to make a page template from where the user can upload the custom profile picture for their buddypress profile.
By default a Buddypress user can update their profile picture at
/members/$username/profile/change-avatar/ - if your theme isn't exposing that option something might be broken in your profile template.
Edit: this functionality is exposed in
/members/single/profile/change-avatar.php, can you just include that in your custom template? If not, examining the functions used on there should get you started.