Does the church of Christ believe that the only way to heaven is through the Lord Jesus Christ, by accepting Him as your Savior and not by good deeds or grace?
"If you love Me, keep My commandments" (John 14:15).
You cannot claim to accept the Lord as your Savior if you are unwilling to obey him. Peter and John stated, "Nor is there salvation in any other, for there is no other name under heaven given among men by which we must be saved" (Acts 4:12). No one else has the authority or ability to save us from our sins. But you can't give only lip service to Christ and expect salvation. "My little children, let us not love in word or in tongue, but in deed and in truth" (I John 3:18). We have salvation because we do the deeds God has laid out for us to do. "For this is the love of God, that we keep His commandments. And His commandments are not burdensome" (I John 5:3). "For we are His workmanship, created in Christ Jesus for good works, which God prepared beforehand that we should walk in them" (Ephesians 2:10).
If I taught that salvation was not by obedience, I would not be speaking the truth of the Gospel. "Do you not know that to whom you present yourselves slaves to obey, you are that one's slaves whom you obey, whether of sin leading to death, or of obedience leading to righteousness. But God be thanked that though you were slaves of sin, yet you obeyed from the heart that form of doctrine to which you were delivered. And having been set free from sin, you became slaves of righteousness" (Romans 6:16-18). The error you are making is assuming that accepting the Lord as your Savior precludes obedience. The truth is that accepting the Lord as your Savior demands that you are obedient to His commands.
But this only speaks of salvation from man's viewpoint. Man is unable to save himself. Though obedience is required by God for salvation, we cannot obey well enough to obtain salvation. Thus the reason we have salvation is because of God's grace. "And you He made alive, who were dead in trespasses and sins, in which you once walked according to the course of this world, according to the prince of the power of the air, the spirit who now works in the sons of disobedience, among whom also we all once conducted ourselves in the lusts of our flesh, fulfilling the desires of the flesh and of the mind, and were by nature children of wrath, just as the others. But God, who is rich in mercy, because of His great love with which He loved us, even when we were dead in trespasses, made us alive together with Christ (by grace you have been saved), and raised us up together, and made us sit together in the heavenly places in Christ Jesus, that in the ages to come He might show the exceeding riches of His grace in His kindness toward us in Christ Jesus. For by grace you have been saved through faith, and that not of yourselves; it is the gift of God, not of works, lest anyone should boast. For we are His workmanship, created in Christ Jesus for good works, which God prepared beforehand that we should walk in them" (Ephesians 2:1-10).
We teach the full truth of God. "For I have not shunned to declare to you the whole counsel of God" (Acts 20:27). If you want to know about salvation, you can't just latch on to a catchphrase. You need to examine all that the Bible has taught on the subject. I would like you to look at all the verses in the chart "Things that Accompany Salvation" and see for yourself what requires of you.