This is EVE, usually there are multiple ways and methods to achieve success. 
Not my forte aside of providing some to prospective buyers but some tend to collect corpses.
Some people collect various types of dolls, apparel and other fashion items, and rare collectible items (from events, related to discontinued features, COSMOS and regular mission items and so on) that are difficult or outright impossible to acquire anymore. 
Personally I tend to collect killrights. 
Another thing some people tend to focus on is collecting killmarks, although this generally has to be accompanied by a corresponding killboard as otherwise nobody can know where the killmarks came from, as farming alts for such is also a possibility.
Thus to have a meaningful high killmark ship you also need to showcase the related PvP prowess, else nobody will take your accomplishment seriously.
Some people do other kinds of collecting, one prominent example being…
Which accomplishment was commemorated by CCP with the addition of an in-game monument.
So as you see people can get creative with their collecting and some cases can put in tremendous effort and ISK or even real-life money doing so.
In the end what matters is to have fun doing so and make the journey worthwhile.
As for specific details as to how to accomplish your collection, it depends on the type, as the examples I’ve listed all require a different method to accomplish.
In general: Be creative, do related reading and research, ask for advice (as you already did) and think about the subject how to accomplish it; and also worth thinking about what other things you might want to collect as it is also an option to seek to accomplish more than one type of collection.