As an Irish citizen, I understand I am entitled to UK citizenship since my grandfather was born in Co. Tyrone No. Ireland in 1897. How can I please go about this process?
You don't have to do anything - you ARE a UK citizen (if your statement about your grandfather entitling you to it is correct - don't know if it's true or not.)
To apply for a UK passport you'll need to supply evidence of that citizenship - ie the appropriate birth certificates of all concerned.
The fact is that the state of Ireland was under British control at that time - all 32 counties - and if you think that makes us all British I'd challenge you to defend your position in any pub in the country...!