Original post by AlexanderHamYou can't seriously aspire to join both. The FBI doesn't take foreign citizens, and both MI5 and the FBI expect you to have lived in the country for the previous five to ten years before joining.
MI5 typically does not use guns; they are not a military or law enforcement agent. I'll explain MI5's division of labour as it might help you to understand the organisation you aspire to join.
First and possibly most important type of intelligence officers at MI5 are case officers. A case officer is a person who goes out into the field to recruit sources. A foreign intelligence (MI6) example will explain it best. If MI6 needs to get hold of a document that they know is sitting in a safe in a building in a foreign country, instead of sending some ninja MI6 officer to break in, they would ordinarily have a case officer recruit sources.
The case officer would try to recruit a cleaner, or someone who works there, to get hold of the document for them. A case officer's bread and butter is being adept at persuasion, having emotional intelligence, understanding what motivates people. Their job is to recruit sources to spy for them, and provide information.
MI5, just like MI6, has case officers and these are the frontline in securing "human intelligence" (intelligence information derived from recruited human sources rather than, say, surveillance). To be a case officer, you will be expected to have reasonably good grades, to have a good understanding of politics and current events, and also to be able to demonstrate a certain comfort and skill in dealing with people. MI5 officers do not go around shooting people or engaging in car chases, it is much more about using intelligence (in the mental sense) and subtlety rather than brute force.
The second group of intelligence officers are the analysts. These are the people who are collating all the information from various sources, including information collected by the case officers from people they recruited, and synthesing this information into analytical products, reports and the like. Analysts need to have good grades and often will be a subject matter expert, for example a financial expert or an Arabic speaker or something like that. They sit behind a desk in headquarters and are often the ones who provide the initial information that allows a case officer to target a potential source.
The third group are the technical/surveilance officers. Despite all the advancements in technology, being able to put someone under physical surveillance and tail them discreetly, is something we still have to do. These guys are the ones who will follow a possible terrorist suspect or suspected Russian intelligence officer. They are also involved in planting listening devices and cameras in the cars, homes and businesses of targets of surveillance. Those people with computer/hacking skills would also come under this group, although a lot of that work is done by GCHQ.
MI5's role is a domestic one. They have no police powers and generally don't operate overseas. They have three main responsibilities. The first is counterterrorism; investigating and surveilling possible terrorists, recruiting sources inside terrorist networks here in the UK, placing agents inside radical mosques or inside far-right nationalist groups. Preventing bombings and attacks in London is a major focus for MI5, and they work closely with the police on that.
MI5's second main focus is counterintelligence. That is detecting and preventing espionage against the United Kingdom by foreign powers. They try to discover foreign intelligence officers in the United Kingdom, and a topical example is that they would be following Russian intelligence officers operating under diplomatic cover at the Embassy, attempting to recruit possibly cleaners or other staff at the embassy, and also investigating penetrations of UK government organisations (for example, if it was suspected Russia had recruited a civil servant at the Cabinet Office, MI5 would investigate).
MI5 also has a responsibility (much reduced since the end of the Cold War) for investigating subversive activity; anarchists, extreme environmentalists and the like. Finally, they provide some support to the police in investigating organised crime where MI5's particular skills and expertise (in human intelligence, in electronic surveillance etc) could be of use to the police.
Intelligence organisations tend to weed out people they think have a "James Bond" mentality, or people going into it for the wrong reason. They tend to recruit people who are astute, sensible and risk averse. They also must recruit people who have unquestioned loyalty to the United Kingdom. Thus, your comment that you'd join either MI5 or the FBI, and your interest in guns, would probably be red flags for them.