Because Ruby: Hard Mode began as Nuzlocke simply making a log of his adventures in Pokémon Ruby, the beginning of this run has little in the way of story. The protagonist, Ruby, simply follows along with the plot of the game, catching Pokémon as he goes.
The first semblance of backstory in the run comes when Ruby challenges his father, the Leader of the Petalburg Gym, as Ruby reveals that the entire reason he began his Pokémon journey was to gain the approval of his father. However, even after Ruby defeats Norman and wins the Balance Badge, the Gym Leader still shows little emotion toward his own son. Because Ruby believes that he needs to justify the deaths of his Pokémon, he continues his journey to become the Champion of the Hoenn League anyway.
Ruby captured many Pokémon throughout his journey, though he restricted himself to only catching the first encounter on each new Route, Town, or Cave he visits.
WARNING: THE FOLLOWING SECTION CONTAINS SPOILERS REGARDING DEATHS.
Treeko/Grovyle/Sceptile: Ruby's starter Pokémon in this game, and also his final death, to Steven's Metagross.
Seedot/Nuzleaf: Ruby's first catch, modeled after John Locke from the TV series Lost. Died to Wattson's Self-Destructing Voltorb, but was never forgotten.
Silcoon/Beautifly: 'Another piece of shit' that Ruby grew attatched to. Poisoned by a Wurmple and died en route to the Pokémon center.
Zigzagoon: Died protecting the weak Whismur from a Poochyena
Whismur/Loudred: Intitially hated by Ruby for causing the death of Zigzagoon, he regained his masters' love because of his immunity to Supersonic attacks. Died to Maxie's Crobat.
Makuhita/Hariyama: One of Ruby's favorites. Died to Steven's Armaldo in the final battle.
'Gulliver' the Wingull/Pelipper: Sacrificed himself to save Ruby while battling Groudon. Sceptile got the revenge kill on the legendary Pokémon.
'Mr. Humpy' the Numel/Camerupt: A female Numel, despite her name. Died to Drake's Salamence, because Ruby was not confident in Katie to defeat the Dragon-type.
Absol: A 'Shadow ballin!' Absol modeled after Soulja Boy, that died while setting up a Perish Song on Steven's Metagross.
'Katie' the Spheal/Sealo/Walerin: Initially a 'derp' that Ruby only added to his team because he needed somebody that could use the Rock Smash and Dive HMs. However, Katie proved her strength by defeating Maxie's higher leveled Crobat, after it killed Loudred. Died to Steven's Metagross.
'Dogen' the Medicham: A philosophical member of the team to replace Gulliver, and the final known catch in the Ruby chapter. He was also Ruby's first Elite Four loss, being one-hit-killed by a Sheer Cold from Glacia's Walerin.
The term "Nuzlocke" originates from the run; it is derived from Ruby's first captured Pokemon, Locke the Nuzleaf. The line "I believe all of this is happening for a reason" is also a staple of the Nuzlocke run.
Ruby: Hard Mode is heavily influenced by Nuzlocke's liking of the TV show LOST.