There are some birth records from Dromore, County Down. If these records belong to the same Ricard Martin, he is the son of Robert Martin and Jane Walsh. He was born on 10 Jan 1856. But I'm not sure this is the correct record for Richard Martin.
What is certain is that he came to the US, married Laura Frances Hubbard, and had a child in 1878. In 1879 he was killed in a rairoad accident - he worked on the railroad. There are a couple of newspaper articles that give the date and a few details.
He had such a short life, and there are very few records for him.
According to family records a brother came from Ireland with him and settled in Monroe Michigan but I have not been able to find any record of this brother and I don't know his name.