How do we know God exists, and that He's Krishna?
To be able to see the Supreme Being personally before you, in His spiritual form, requires great qualification. Just as you can't demand the president of a country to appear before you, you can't demand that God appears before you.
Many people throughout history have independently realized and confirmed Krishna's existence and His identity as the Supreme Person.
By following Krishna's instructions in Bhagavad-gita, anyone can see how their life improves.
In the Bhagavad-gita, Krishna thoroughly defines the Supreme Person—and acknowledges that He Himself is that Person—in such a way that any impartial, reasonable reader would be convinced that He is, in fact, God.
"I am the source of all spiritual and material worlds. Everything emanates from Me." (Bg. 10.8).
"I am seated in everyone's heart and from Me come remembrance, knowledge, and forgetfulness." (Bg. 15.15).
If someone genuinely wants to know the identity of the Supreme Person, the above statements will be helpful signposts on the path to God-realization. On the other hand, if someone wishes to avoid the question of God/Krishna's existence entirely, no amount of logic, practical advice, or scriptural reference will be sufficient to convince them either that God exists or that God is Krishna. Krishna is a Person, and His energies work in such a way as to bewilder atheists and attract His devotees.