(Clinton laughs, thinking she'll win, and wants to "get down to it", meaning to get it over with since she wants to finish the battle already.)
You done messed with the wrong Rodham. I'm about to Tudor and boot it!
(Clinton's maiden name is Rodham, and she tells King Henry not to challenge her. The term "Tudor" refers to the English royal dynasty who ruled between 1485 and 1603, which King Henry was a member of. It may also be playing on the word tutor, in which case she is also saying she will teach him a lesson. "Toot It And Boot It" is a 2010 rap song by YG about having sex with women, and then immediately "booting" them out of your house, which is something that King Henry VIII has been guilty of many times.)
Your country's the size of Ohio. You couldn't even beat Spain!
(Ohio has been one of the most important swing states in the US during the 21st Century, and it is one that Clinton won in the 2008 Democratic primaries. Ohio is the 34th largest state of the 50 states of the United States of America, therefore Clinton tells King Henry that his country, England, isn't very large compared to the USA, so his leadership of it is less impressive than Clinton being the Secretary of State of, and potentially leading, the USA. King Henry also battled Spain and lost.)
I could defeat you with a grassroots email campaign!
(A grassroots campaign is a political movement driven by a community, in this case via email. Clinton has also implemented grassroots email campaigns in 2008 and plans to implement it again for 2016, when she runs for President. Clinton says she could easily beat Henry using this method.)
I'd eat you alive…
(Clinton is depicted as an independent/powerful female icon, so she says she would be able to defeat King Henry with ease. To "eat someone alive" in a figurative sense means to overwhelm or defeat someone, often inexperienced, thoroughly. This also leads up to another diss, as this is where the snippet ends.)