The method I've been going with has been:
1. Go do a regular VS mode. Make things quick with 1 stock setting.
2. Use the character that you want customs for and play against any cpu. I recommend winning.
3. Go to Target Blast, stage 1, mainly because it's the easiest stage, and pick, again, the character you want customs for.
4. The tedious part is keep resetting the map until the green customs target is spawned.
5. Play Target Smash as normal, aiming for the custom part, and you'll get your reward.
Rinse and repeat steps 4-5
Supposedly, after reading up reddit, the first game of Target Smash that you finish has the highest possibility of rewarding you with a custom part (but not guaranteed to be ones that you want). Again, it's still all RNG from there. However you are currently swaying the RNG to give you parts towards the character that you are using.
I do this method because I can keep track of the customs that I get. Plus, I don't have a turbo controller

If that sounds somewhat tedious you can try the other method that Deviant posted.