Luke 22:3 and John 13:27 mention the point in time when Satan entered Judas Iscariot. Should Judas be held responsible for betraying Christ, or Satan within him?

The initial impact of Satan upon Judas is told us in John’s gospel: The evening meal was being served, and the devil had already prompted Judas Iscariot, son of Simon, to betray Jesus—John 13:2. The idea in the Greek (the word used here is ballo (bal-lo) meaning to cast or throw, when used with eis (ice), meaning “into,” is metaphorically meaning to put into one’s heart, suggest to one’s mind) is for someone to throw an idea into someone’s heart or mind. Later on, the interaction with Satan becomes much stronger. As Luke tells us, Then Satan entered Judas, called Iscariot, one of the Twelve—Luke 22:3. John’s gospel extends this idea by telling us that, [a]s soon as Judas took the bread, Satan entered into him. “What you are about to do, do quickly,” Jesus told him, but no one at the meal understood why Jesus said this to him—John 13:27-28. A word from the Greek helps us to understand exactly what happened with Judas: eiserchomai (ice-er^-khom-ahee), which means to come in, come into, enter in or into, go in or through. Judas, his life never surrendered to Christ anyway, had now become possessed by Satan in his plans to have Jesus arrested. Whatever Judas’ ulterior motives, he was now fully Satan’s agent in the devil’s attempts to derail God’s plans through His Son.
What is also important to remember is that, regardless of how one views Satan’s influence over Judas’ choices, in the final analysis, God, in the person of Jesus Christ, was the one who moved every participant’s actions forward toward an end that would accomplish God’s plan and give God all the glory. As Peter tells us, For you know that it was not with perishable things such as silver or gold that you were redeemed from the empty way of life handed down to you from your forefathers, but with the precious blood of Christ, a lamb without blemish or defect. He [Jesus] was chosen before the creation of the world, but was revealed in these last times for your sake. Through him you believe in God, who raised him from the dead and glorified him, and so your faith and hope are in God—I Peter 1:18-21. Jesus Himself during His time on earth declared in no uncertain terms just Who was in charge during those final critical moments of His earthly life, as He said at one point to the Pharisees, The reason my Father loves me is that I lay down my life—only to take it up again. No one takes it from me, but I lay it down of my own accord. I have authority to lay it down and authority to take it up again. This command I received from my Father”—Jesus Christ, John 10:17-18. And, as He said to Pilate just before His crucifixion, You would have no power over me if it were not given to you from above—Jesus Christ, John 19:11a. No one, including Satan, exerts power or influence over the Father. As He memorably said through the prophet Isaiah, I make known the end from the beginning, from ancient times, what is still to come. I say: ‘My purpose will stand, and I will do all that I please’—God speaking, Isaiah 46:10.
Regarding who is ultimately at fault for the betrayal of Christ—Satan or Judas—Jesus Himself gave answer when He said regarding Judas, The Son of Man will go just as it is written about him. But woe to that man who betrays the Son of Man! It would be better for him if he had not been born—Jesus Christ, Matthew 26:24. Clearly, Judas is judged as guilty for his actions. Yet, just as clearly, Satan was permitted to have a profound influence on him. And again as clearly, God allowed both individuals to play out their planned actions—which were most fundamentally His planned actions. The Bible also teaches, as Jesus said, No one can come to me unless the Father who sent me draws him, and I will raise him up at the last day—Jesus Christ, John 6:44. The Bible tells us that [N]o one can say, “Jesus is Lord,” except by the Holy Spirit—Paul, I Corinthians 12:3b. And, as Jesus Himself so simply and clearly said, [A]part from Me you can do nothing—Jesus Christ, John 15:5b. What it all ultimately means is a mystery to us, but it makes perfect sense to God. God is completely sovereign and yet, at the same time, people will be judged as rejecting Christ and as worthy of hell—judged absolutely perfectly by God Himself. Completely explaining an infinite God to a finite mind will never be possible, which is why we rest in the truth of the love and grace of God of which the Bible tells us.