Can I be forgiven my sins

by doing good works?

No. The Bible says, “But we are all as an unclean thing, and all our righteousnesses are as filthy rags…” (Isaiah 64:6). God rejects any attempt to earn our own salvation: “For by grace are ye saved through faith; and that not of yourselves: it is the gift of God: Not of works, lest any man should boast” (Eph. 2:8-9). God wants to freely give us salvation and to clothe us in His own righteousness. The person who works for his or her own righteousness with an “I can do it myself” attitude is highly offensive to God.