Press Down, C, Up, A(2), B(2), C when the building at the start of the introduction sequence fades. Alternatively, use a six-button controller and press Down, Z, Up, X, A, Y, B, C at the Street Fighter 2 logo.
Hold Start after character selection. A controller configuration screen will appear.
If someone (Dhalsim maybe) keeps beating you select a strong player at the start of the game.
Then once you've beaten the character that you usually can't beat let the person after that defeat you to get to the player selection screen. Don't press [Spacebar] but with the cursor keys select the player that you can't complete the game with.
You can now finish off the rest of the characters and complete the game.