How can you justify Paul's claim as being an apostle if he is a liar, who spends a lot of time claiming he isn't a liar? Shouldn't one use other sources to substantiate a person's claims? If I used my wife or children as an alibi, the police would look for someone outside my family or close-knit friends to verify my story of where I really was a so-and-so time. So how can you use an accused man's own words to vindicate him?
"And consider that the longsuffering of our Lord is salvation -- as also our beloved brother Paul, according to the wisdom given to him, has written to you, as also in all his epistles, speaking in them of these things, in which are some things hard to understand, which untaught and unstable people twist to their own destruction, as they do also the rest of the Scriptures" (II Peter 3:15-16).
Peter's testimony is that what Paul wrote is Scripture -- sacred writings inspired by God.
The book of Acts is written by Luke, the same author as the book of Luke. Luke is a meticulous historian documenting both Jesus' life and the early church. Luke also testifies to Paul's conversion and his commission from Jesus, citing people like Ananias to validate the events (Acts 9:13-15). Luke also documented the miracles that were done through Paul. "This man heard Paul speaking. Paul, observing him intently and seeing that he had faith to be healed, said with a loud voice, "Stand up straight on your feet!" And he leaped and walked" (Acts 14:9-10).
Since miracles are from God, this tells us that God also testified to Paul's authority as an apostle. "God also bearing witness both with signs and wonders, with various miracles, and gifts of the Holy Spirit, according to His own will?" (Hebrews 2:4).
Paul points out that the Thessalonians he wrote to know about his apostleship, so we can add the churches who received Paul's letters and who distributed them to all other Christians as supporting evidence of acceptance of Paul's claim. "For you yourselves know, brethren, that our coming to you was not in vain. But even after we had suffered before and were spitefully treated at Philippi, as you know, we were bold in our God to speak to you the gospel of God in much conflict. For our exhortation did not come from error or uncleanness, nor was it in deceit. But as we have been approved by God to be entrusted with the gospel, even so we speak, not as pleasing men, but God who tests our hearts. For neither at any time did we use flattering words, as you know, nor a cloak for covetousness--God is witness. Nor did we seek glory from men, either from you or from others, when we might have made demands as apostles of Christ. But we were gentle among you, just as a nursing mother cherishes her own children. So, affectionately longing for you, we were well pleased to impart to you not only the gospel of God, but also our own lives, because you had become dear to us" (I Thessalonians 2:1-8).
The number of witnesses is sufficiently high enough that we can accept Paul's own words on the matter, especially knowing that his writings were from the Holy Spirit.