Are there any example of runic writing of a foreign name? Maybe journals or histories from famous warriors, travelers or foes?
If so, how they do it with the runes?
Basically I'm trying to find how would they write my name
This is one of the medieval rune inscriptions from Bryggen in Bergen, Norway. See https://en.wikipedia.org/wiki/Bryggen_inscriptions. It is in Latin and reads "rexiudeorum innomini patrisnazarenus" - "Rex Judæorum In nomine Patris Nazarenus" - "King of the Jews in the name of the Father of Nazareth".
There are also plenty of foreign names in inscription B013, same wikipedia page.
The answer to the OP's question would seem to be "straight transliteration". "Mark" is easily expressed in all of older, younger or medieval runes.
According to Runes and Runic Inscriptions: Collected Essays on Anglo-Saxon and Viking Runes By Raymond Ian Page
The vikings used feminine gender (irregular weak feminine*).