In the beginning, they got along all right. She was his little baby sister, only a few years younger to him, but nonetheless, he felt a sort of protective instinct towards her. He would watch over her whenever his parents let him take the "responsibility" of doing so, and would do it with pride. That only lasted so long, however. More specifically, until she was old enough to show him she was quite responsible herself, thank you.
They were siblings, after all, and that fact manifested as they grew older. It was terribly annoying to him that she, by virtue of being younger and cleverer, often got a huge advantage against him. Amycus didn't like the idea of not being the center of attention anymore, and so their relationship turned into some not-so-friendly sibling rivalry. Their parents hardly seemed to even care, so they were often found to be engaging in nearly warlike skirmishes.
It only grew worse once they were both sorted into the same house at Hogwarts, and Amycus tried to sway the Slytherin crowd to his side, given that he had a few years' head-start. Nonetheless, he didn't quite succeed in his endeavour. Alecto excelled at school, easily winning over both her peers and her teachers. Amycus often ended up resenting having a sister after all.
Once they had both joined the Death Eaters, they kept up their tradition of trying to out do each other. This sometimes led to quite a few more Muggle deaths than was needed, which in turn, pleased the Dark Lord greatly. They quickly rose to the top of their ranks, becoming among Lord Voldemort's favorites. This was why, when after the siege of Hogwarts, they were sent to teach there, both siblings were ecstatic. It was another chance for them to prove who was better. Of course, this meant that the students suffered the brunt of their dislike for each other. The children would often find themselves caught between the two teachers who were trying to make their punishments successively worse. Occasionally, Snape had to step in to prevent them from permanently damaging the students in the process.
It was at the Battle of Hogwarts that this old feud was finally laid to rest. Having thrown a few curses and hexes at the nearby students, Amycus was trying to defend himself, when he noticed his sister's absence. Only a few moments ago she had been zealously fighting, and it was unusual for her to simply vanish. She was hardly the type to run and hide, he knew. He then saw her limp figure lying amidst the rubble, some distance away, and instinctively ran towards her. He was lying there, motionlessly with her eyes wide open and and expression of pain on her face. The pallor of her skin and the state of her body made him tremble with fear as he kneeled down beside her and tried to find her pulse, but to no avail.
He was still sobbing when they led him away to Azkaban. No one knew the pain he felt, the pain of realising many years too late that he did love his sister after all.
Notes: For the TGS Scavenger Hunt!