How to activate my iPhone 5?

Activation Lock?, See this Apple article: How to remove Activation Lock

How to remove Activation Lock - Apple Support

Have you forgotten your Apple ID?, check your Apple ID via your email box, see this Apple article: If you forgot your Apple ID

If you forgot your Apple ID email address or phone number - Apple Support

OR see -> Having trouble signing in?

https://iforgot.apple.com/password/verify/appleid

Then see this Apple article: What to do after you change your Apple ID or password

What to do after you change your Apple ID or password - Apple Support