Whether it be getting the Elite rank through combat, trade or exploration, this trophy takes a sizeable amount of grinding. As stated elsewhere, the benchmarks have changed over time on the other formats, and there was no fixed figure. It is better to judge your progress it by your own stat figures. (NB some of the information here presumes you are not an absolute beginner, e.g. you know how to hire crew, you can do a galaxy map search for a system in game, you know how to do planetary scans, you understand the expression 'going out into the black')
NB2: This game is prone to changes with patches, specifically nerfing of any methods the developer perceives as to exploitative of a way of making money. Optimistically the information below will be future-proof and useful to you.
The most 'basic' of three avenues is trade. Get the bulkiest ship you can buy (ie. lots optional internals for cargo), and make use of eddb's loop route finder
https://eddb.io/trade/loops (thanks to the creaters and contributors on eddb.io and 'ross')
Be careful of how old trade data is and adjust your search if necessary. Obviously be wary of landing pad size. Also, Imperial slaves is usually one of the two commodities on the list, a decent amount of pre-existing capital will be needed. If you don't have capital yet, I recommend either a small bit of exploration (see below), or some mining (if you havn't already got the Refined Taste trophy), off of Delkar 7 is my recommendation.
This is IMO the grindiest and most boring of the three Elite ranks to get, even with a huge cargo ship. Furthermore, if you rinse a single supply of a commodity too heavily, you can actually alter the offered prices, so variety in trade routes needs to be considered.
The trade ranks (so you know where you stand) are:
PENNILESS
MOSTLY PENNILESS
PEDDLER
DEALER
MERCHANT
BROKER
ENTREPRENEUR
TYCOON
ELITE: A conservative estimate is currently over 1 billion credits in your 'trading profit' stat.
Combat is more fun, but the sheer number of kills means it will take you a long time to get to Elite, however good you are (
http://elite-dangerous.wikia.com/wiki/Combat puts it at roughly 2,400 average kills just to go from 'Deadly' to 'Elite' ).
If you want a lazy way to get kills fairly early on, get a keelback, a fighter hanger, and half way decent ai crew. In 1 on 1 combat situations set your crew flown fighter on the enemy first of all, and then with that distraction ongoing, you should have free reign to take out the distracted ship. My personal preferences for weapons are burst lasers, multi-cannons and seeker missiles, but everyone has their favourites.
The tried and tested method of getting kills, going to mining 'zones' works (where you ride the bandwagon of security forces taking out pirates, making sure you get the killshot for the 'glory'), but when you end up in more heavy duty ships, the high threat/security combat zones might be more consistent for getting lots of kills.
The combat ranks (so you know where you stand) are:
Harmless
Mostly Harmless
Novice
Competent
Expert
Master
Dangerous
Deadly
Elite
At face value, exploration seems as much of a grind as trade, if you are deliberately trying to get Elite rank. You rank up by selling exploration data to Universal Cartographics at space stations. Data is gathered by use of Discovery scanners and surface scanners.
There is a way to reduce the grind, if that too is a bit grindy. If you want to get Elite rank in exploration I recommend you pledge and get rating 5 with Li Yong-Rui's Faction in Galactic Powers - Power Play. Getting this rating will give you a 200% increase in the money given for sale of exploration data to universal cartographics, as long as the sale is in within a Li Yong-Rui controlled/exploited system. If you want to go from 0 to 5 rating in a single week, you can bypass the cooldowns for prep/expansion/control units, but it will cost you near 100,000,000Cr and roughly 40+ trips even with a large cargo ship.
Whilst it seems like a big hurdle, the tripling of sales value of exploration data is worth it.
The next step I would recommend is use of eddb.io again:
https://eddb.io/body (as ever thanks to the contributors to on eddb.io and 'ross')
Use that page to refine your search for systems near you, preferably uninhabited. Using this chart as a reference,
https://postimg.org/image/lquhdv87d/ (many, many thanks to Cmdr Fru for this)
If you look at that chart you can see earth like worlds, water worlds, and ammonia worlds give the biggest pay out when scanned by a surface scanner. high metal content worlds shouldn't necessarily be ignored, but they're terraformability is a gamble. I would go 'out into the black' from the Li Yong-Rui controlled systems, then use eddb to search for uninhabited systems with the above planet types, and go round the houses so to speak, before heading back into cash in my exploration data after an hour or two (you don't really want to risk getting interdicted by an Anaconda with over 30 mil in exploration data).
I'd recommend either a Diamondback Explorer or AspExplorer for this job, streamlined for weight, fit with engineered 5A frame shift drives for longer jump ranges, and the best fuel scoop you can grab.
Worth noting too is you can get a 50% bonus on scans of stars and planets if you are the first person to scan them, however you have to travel quite a far distance from 'the bubble' (inhabited systems) before these crop up.
If you do go out into the black, it's worth refining your galactic map to only show scoopable star systems (KGBFOAM is the acronym to remember via the star class option on the map filters tab). This prevents any risk you might become stranded.
The Exploration ranks (so you know where you stand) are:
Aimless
Mostly Aimless
Scout
Surveyor
Trailblazer
Pathfinder
Ranger
Pioneer
Elite: I got mine breaking 330,000,000Cr in 'profits from exploration', and it seems getting over the 300 mil mark is a consistent benchmark at this point in time.
Just to make things a little easier, here are a couple of systems worth scanning:
Col 285 sector AB-O D6-107
Col 285 Sector CX-J C9-8
Brib
ps thanks to Ziggaman on trueachievements for putting all the rank titles in a single place.