During my play through of Fire Emblem: Awakening I came across a character that was a recruit able but she ended up being taken out through some unfortunate circumstances. In case it is relevant it was Tharja during a sequence in Chapter 9. Is there any way for me to recruit this (or any missed character) through some other means? Or if I miss the opportunity is it impossible for me to get that character through normal means?
Sadly, those characters only give you one chance to recruit them. If you don't have the correct person talk to them, then you can't recruit them.
Note that this is true even in Casual mode; Casual only flags your characters as unable to die. Until you recruit them, you can definitely kill them, so be careful if you're trying to get everybody.
One thing to do at the beginning of each mission is to run through every enemy; a recruitable characters will have a name and a unique portrait, and not be the boss. Bosses cannot be recruited except through story means, so there's no need to be careful with them.
When you find a character you can recruit, tap their name; it will often give you a hint as to who needs to talk to them to recruit them. Usually it's Chrom, but it might be someone else.
Awakening also has a neat feature where if the person who can talk to them is within movement range, they'll have a "Talk!" icon above their head when you try to move that person. That's new to Fire Emblem; previous versions meant you had to move a person beside the recruitee, and see if a Talk option was available. If not, cancel, and choose the next character. That still works in Awakening, but they make it easy on you with the icon.
No. There is no way to recruit characters you missed at a later time.