My current class quest for archer is reaching 150 fame with the archer character. Will this class quest prevent me from earning more than 150 fame with my current run?
No it won't.
Your fame is counted in total and you can go over to achieve other things.
150 fame is one of the 5 class quests (The 5th one being 2000 fame). You can earn an unlimited amount of fame with a character, and the class quest fame does not have to be base fame (fame earned while alive). Completing class quests gives you stars. More info on that and class quests can be found here
To answer your question, 150 fame is not the max.
Once your archer gets to 150 base fame, it will say Class Quest Completed. Your fame bar will become bigger, showing that your next class quest is 400 fame. You can keep earning as much fame as you can. If you complete that, it goes to 800 fame and then 2000 fame. After 2000, Your fame bar simply shows the number of fame you have (Ex. 300 Fame instead of 300/400 Fame).
Simple answer: No fame limit.
When your character reaches 150 fame, a new goal will be set.