Strength
If there was one thing that Bellamy had always possessed it was strength. Ever since Octavia had been born that's what Bellamy had been forced to have. He could never have fun like a normal kid; he always had to take care of his sister and put her needs and wants before his own. Most days this fact didn't bother him and he lived for the moments when Octavia could come out and he could make her laugh. To Bellamy, nothing was more important than Octavia's safety, and he had learned early on that the best way he could insure that was to be strong enough for the both of them.
After coming to Earth, Bellamy didn't think anything would change; he would continue to be strong enough to protect Octavia and himself and everything else would fall into place. But the situation quickly got out of hand and before he knew it he suddenly had to be strong enough to lead and protect this band of misfits he had unknowingly joined. They were facing a war that would probably lead to a majority of their deaths, because despite all of their advanced technology, the Grounders knew this land and had a savagery that dwarfed anything the 100 were capable of. They may all be delinquents, but they hadn't been raised to be the kind of warriors the Grounders molded their children into. So now, instead of being strong enough for two people, he had to be strong enough for 100.
Back on the Ark, Clarke had learned, as soon as she found out what her father had discovered about the Ark, what it meant to be strong. Watching her dad struggle with the decision of what to do with what he knew and eventually deciding to expose the truth, she learned the strength it took to shoulder the weight of a life-altering decision. After being accused of treason and thrown into solitary, Clarke learned a different kind of strength. She learned the strength it took to rebuild the foundations of her life after they had crumbled beneath her. That strength, forged by hours filled with only her own thoughts and sketches to keep her company, was the armor she used to keep herself safe from the pain of her father's death.
On the ground, that strength she had only ever used to protect herself quickly changed to include all of the 100. As the only one who had any sort of medical training, Clarke had to adjust to being needed by so many people. Even before when she had been training with her mother, Clarke hadn't been the one the patients relied on; that job had been filled by Abby Griffin. Besides gaining the strength of a doctor, the gentle strength that reassured patients and didn't waver no matter what, she also adopted the strength of a leader. There were some things she just couldn't stand by and let happen, and before she knew it Clarke had established herself as a leader of this band of rebels who dared to live.
Both Bellamy and Clarke had been strong and knew strength before coming to the ground, but it was Earth that taught them their true strength. By establishing themselves as co-leaders, they unknowingly opened themselves up to the pressures and weights that came with trying to keep people alive in a situation that was trying its best to kill them. Individually, they were stronger than most could ever dream of being, but together they seemed invincible. With Bellamy's push towards structure and iron-clad leadership and Clarke's belief in people's better natures, they butted heads more often than not, but when they did come to an agreement, it was apparent to everyone that this decision was the best option they could hope for. Once they had learned to trust each other and go to each other before making decisions, their strength gave people hope that maybe they could survive this war with the Grounders.
Weakness
The lives Bellamy Blake and Clarke Griffin lead left little room for weakness. They were the ones who always had to be strong for others, to help inspire courage and hope in the people who looked to them for guidance. In this world they found themselves thrust into there was no room for weakness. This world killed the weak and that was a simple truth. But Bellamy Blake and Clarke Griffin were also human and that simple truth meant they couldn't be strong all the time. The weight they had been forced to take on was part of a herculean burden they couldn't carry every moment of every day. Being who they were though, neither one was willing to admit that they needed a break from the stress.
Gradually, the two leaders realized they didn't have to be strong in front of each other. As their working relationship grew and settled into a pattern, their personal relationship went from straight out animosity to grudging respect to a fondness and reliance that neither would admit to easily. After Dax's attack, Bellamy was the first to show the weakness and doubts he kept from everyone else and Clarke was there, supporting him and helping him pick up the pieces. Bellamy returned the favor after the Exodus ship crashed and Clarke couldn't hold onto her strong façade in the face of the death of her only parent left. They were there for each other when the strain became too much.
Eventually, they developed a habit of going to each other even when they hadn't reached their breaking points but simply because they wanted to relax. All the pretenses they had to keep up for the others melted away when they were alone. These times were filled with easy silences or, if they felt like talking, pointless conversations full of sarcasm and biting wit that had long ago lost its actual sting. They might get along much better, but Bellamy and Clarke were still two very different people who genuinely enjoyed poking fun at the other just to see what reactions they could get. The comradery between the two acted as a support beam, something they could rely on when the world threw too much at them at once. There was comfort to the openness they shared together, how they didn't have to be anyone but themselves, one that made everything a little easier.
Bellamy Blake and Clarke Griffin both had immense strength, but it was the weakness they only showed to each other that truly bonded them.