Counting in bijective base 62
Jelly, 7 bytes

ØWṖṗR;/


This is a monadic link that accepts an integer as input and returns an array of strings.

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How it works

ØWṖṗR;/  Main link. Argument: n

ØW       Yield 'ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789_'.
  Ṗ      Remove the last element (underscore).
    R    Range; yield [1, ..., n].
   ṗ     Cartesian product. For each k in the range, this yields the arrays of all
         strings of alphanumeric characters.
     ;/  Concatenate the arrays of strings of each length.