Wikipedia has an article with a section specifically about the historicity of the Abgar legend (emphasis mine):
The Abgar legend has played an important part in the self-definition of several Eastern churches, but its historicity is extremely doubtful. Two recent histories of the Church of the East, Baum and Winkler's The Church of the East and David Wilmshurst's The Martyred Church, have addressed this issue and have discussed the growth and development of the legend. Alexander Mirkovic also argued against the historicity of the legend, pointing out at that the legend is not the only one of its genre. There were many conversion stories coming out of the Middle East in the 3rd and 4th centuries. In many ways these stories represent a model for the conversion of Constantine. (Notice the similarity between the Book of Abgar and the conversion of Helena of Adiabene and her son Ezad II in Flavius Josephus: Jewish Antiquities, XX 2. In the story told by Josephus there is a Jewish merchant by the name Ananias and the question is of circumcision of Ezad.)
In the Book of Abgar, Ananias is the messenger sent by King Abgar to Jesus. Ezad’s son was Abgar VII of Edessa (Ostroene). The origin of the story may be that Ezad, the father of Abgar VII, had exchanged letters with somebody in Jerusalem, but more probably with the Nasi Gamaliel than with Jesus.
To this I'd add that the text of the supposed letter from Jesus seems suspiciously pseudepigraphical. I've highlighted three phrases that don't ring true for me:
Blessed art thou that hast believed in me, not having seen me.
For it is written concerning me that they that have seen me shall not believe in me, and that they that have not seen me shall believe and live. But concerning that which thou hast written to me, to come unto thee; it must needs be that I fulfil all things for the which I was sent here, and after fulfilling them should then be taken up unto him that sent me.
And when I am taken up, I will send thee one of my disciples, to heal thine affliction and give life to thee and them that are with thee.
"that hast believed in me, not having seen me" seems to be an allusion to John 20:29*, which is odd, because the letter would have had to be written before the events described there.
"be taken up" seems to be a direct allusion to the ascension of Jesus. Although Jesus regularly alluded to and predicted his death and resurrection, including the manner of it, he did not clearly state that he would be taken up in the way later described in Acts 1. This again makes the letter look looks like a pseudepigraphical later writing drawing on works written after Jesus death and the completion of the 'canonical' writings.
"I will send thee one of my disciples" simply seems out of character. Jesus had no need to send anyone in order to perform a healing, and the claimed journey of a disciple to Edessa has the ring of a convenient divine endorsement for Abgar's rule there.
* Or possibly to 1 Peter 1:8