Credits to MegaManSurvives from TrueAchievements.
To beat the final boss use the catapillars to refill on lives, health, health in your Sub-Tanks, and weapon power. To make this go fast just use a fully charged Rolling Shield and stay where the catapillars are as HolyRabbitThug stated.
Hopefully you have all upgrades at this point, if so this fight is a lot easier.
#1 Sigma's Dog: Use the Hadoken (I find it easiest to use the control stick for the motion) on Sigma's Dog. Do not move, and do not have a charged up shot as that will hit before your Hadoken does giving him invincibility frames to avoid the Haydoken.
#2 Sigma with Sword: Hadoken him right off the bat, if done correctly you should be at Sigma's final form (his Wolf Face).
#3 Wolf Sigma: He is immune to Hadoken in this form so here's some tips. Only fully Charged X-Buster shots and Armadillo Shield will hurt him. There's a spot on his platforms that you can stand with one foot in the air and 1 foot on the pad. He can only hurt you from here when his other platform runs into you. Use the Chameleon Invincibility fully charge power to get to this position safely.
Once you beat Sigma's final form, watch the ending, and let it roll through the credits you will come to a picture that thanks Capcom on it and it will remain there for about 30 seconds to a minute. You don't need to press anything just let it continue and a secret message from Sigma will pop up. As soon as that happens the achievement will pop.
If I left anything out or you know something that help just message me so we can have the best information available, thanks!