I didn't accomplish "Find out why Tychus Findlay is always in a Marine suite" achievement. If I start the campaign again will I be able to get it?
Yes, you need to start the campaign again though, you can't get it through the archives, as you don't have the post-mission chat with the characters this way.
There are also multiple achievements that require some dialog with the characters; if you don't care about listening to them, at least make sure you click on them when you return from every mission, and just click repeatedly to skip the conversations.
You can also see which character "has something to say" as their name appears with an arrow pointing at them.