To unlock three of those characters, you need to unlock the Young Indy bonus mission. Complete any nine artifacts (find all 10 treasures in any 9 of the 18 missions) then go to Indy's office and:
1. Use Willie or an explosive to shatter the glass in the corner to reveal a chest.
2. Use a lady to jump onto the shelves in the back to reveal another chest.
3. Have a scholar decipher the heiroglyphics in the back left to reveal another chest.
4. Blow away the shiny discs on the left with an explosive and dig to reveal another chest.
5. Use the Thuggee statue on the right, then crawl into the resulting vent to reveal another chest.
6. Assemble the pieces and smash the resulting box to reveal a key.
7. Take the key into the artifact room, then take the blue diamond in the middle of the room and put it onto the pedestal to reveal a crank.
8. Put the key into the crank and turn it to open up the Young Indy mission.
9. Complete the Young Indy mission to unlock two more characters and open up a third to buy.

To find the fourth, you need to find all five Star Wars cameos:
1. In the first mission in Raiders, just after snagging the idol, fire an explosive at the gray wall to open up a new room, then smash the blocks in the new room.
2. In the second mission in Raiders, in the area with the collapsing ice bridges, fall down the third one, then blast or scream at the wall of ice in the back.
3. In the third mission in Raiders, in the last room with the truck, decipher the hieroglyphics at the back wall to open up a new room, then crawl through the vent in the back.
4. In the fourth mission in Temple of Doom, in the fourth area, go towards the right edge to see some railings that slope down. Drop down to the right, blast the rubble, and go through the soldier door.
5. In the fifth mission in The Last Crusade, in the first area, go to the left side and use the bucking horse to get up to the top, then head left and use the whip platform.

Do all of those and you'll open up the final character for purchase.
(The last two items in the character select screen are your custom characters.)

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