it's kind of self explanatory, but almost bestie is someone who is not quite your best friend but is nearly.
It is also a ironic expression for bagging your best friend by sayin that they are only your "almost bestie".
claire: "helen, fuck off, you are only ,my almost bestie!!!"
helen: "(sob, sob) i thought you were my best friend"
helen: "you just failed me by not giving me your lunch, you are no longer my almost bestie! (laughs hilariously)
claire:i didn't want to be you stinking almost bestie anyway, i've got my friend amy instead!"