Some of the earlier posts here nailed it. Firstly don't forward settle Elizabeth if you want to be friends - In fact this goes for any Civ except the most peaceful personalities like Babylon and Brazil.... you can get away with it with Civs like these but you will still have penalties, i.e there will be a period where they will refuse a DoF...
If you do forward settle a Civ usually that tells them to start preparing for war and they will immediately start building armies to attack you (again the peaceful Civs like Brazil may be the exceptions to this). If you do this to Elizabeth then that will sour things greatly and will probably make her "Covet your lands". If that happens it is probably impossible to have a DoF, maybe even until you can share ideologies. Forward settling just has a very strong diplomatic penalty attached to it - so you've been warned...
But what you can also do to become friends is to tick as many of these boxes as possible...
-Share religions (this is very powerful in the early game)
-Pay tribute - give England a luxury for free or 5 gpt... (this is good but I find the effect can erode quickly if the Civ is generally hostile)
-trade items and send a trade route to her cities (I believe that a trade route has a hidden positive effect, after all it benefits the other Civ so it should make them less likely to attack you)
-Liberate her workers if they have been captured by barbarians or liberate cities if they have been captured by another Civ...
-Create an archaeological site in her territory
-Have a medium sized army
-Try and have a marketplace, Colossus or East India Company in a city that her cargo ships and caravans will reach. Again I'm not sure of this but if a Civ is sending you a trade route than that may make them like you even more.... try to get it in a city that has a river and has completely different luxury and strategic resources to what Elizabeth's cities have. The more different resources the more valuable the trade route is.
-Don't spy on her (if a spy is caught, even if you promise not to spy again the damage is done and you end up with a near permanent negative mark). If you can spy on one of her enemies, there is a chance you can gather intrigue (e.g Shaka is plotting against Elizabeth) you can share that information with England. Also the diplomatic benefit of share intrigue stacks, so if your spies give you 2 sources of intrigue against the one Civ in a short period of time it becomes a bright green positive (instead of the lighter green)....
-Vote for the same WC resolutions or if you don't have many WC votes you can vote for England to move the WC to her capital. This is situational though, don't vote for another Civ unless you are friends with them and voting for them will make them win the WC or possibly come second. If your smart you can sometimes displace a hostile Civ from getting the WC who may intend to propose resolutions that harm you. A friendly Civ for instance won't propose resolutions that negatively affect you...
-denounce the same leaders etc etc.....
-Lastly of course you need to share ideologies. Ideologies seem to be the strongest diplomatic modifier in the late game (like religion in the early game). If you end up with different ideologies it becomes very hard to maintain good relations...
On a side note I'm not sure about this one but Elizabeth's UUs come in the mid game. If the devs have coded her to be aggressive it is most likely to occur then. Once you see she has Longbowmen try and get her to go to war with someone. If both of you agree to attack a Civ that's even better as you get the "We've fought together against a common foe"....
Generally speaking most Civs that have warmonger attributes seem content with being at war with only 1 civ at a time. So if England and you are at war with another player than there should be almost no chance she would backstab you...
