How can MRSA get in the bloodstream?

Line or skin. Mrsa (like all staph aureus) lives on the skin, so anything that breaks the integrity of the skin (be it a cut/scrape or an IV line) can introduce it into the bloodstream. This also includes direct injection, such as in IV drug use. Finally, a more rare cause would be a severe alternative infection focus, such as pneumonia.
From skin. Mrsa is a type of staff aureus bacteria that is resistant to the penicillin type antibiotics. Staff aureus lives on everyone and usually just hangs out on the skin and nice warm moist spots like groin arm pits nose if the bacteria gets access to the blood stream like a bad skin infection or less commonly a cut on the skin it can get to blood stream and cause a blood infection.