I suggest reading accounts of real sieges first of all. I read the siege of Lisbon in 1384, written by the chronicler Fernão Lopes, and it gives a vivid image.
The way to portray a siege depends, in my opinion, on the POV: is it a high class person or low class?
These are the people who suffer first. If they lived outside the castle and hurried in to save their lives, then they'll be suffering over the lost lands and animals, probably feeling they've just lost years of hard work. They'll enter the town in family groups with a few essentials and will face the hardship of living in the streets.
Once the gates are locked, the food prices will start climbing and the poor people will be the ones to first experience hunger. As the siege continues, hunger will climb up the social ladder.
In 14th century Lisbon, the food shortage was such that people killed all horses, mules and oxen for food, once the pigs and chicken were gone. Children and teenagers would roam the streets in search of rats and birds. Never had the city been so free of vermin! At a certain time, there was a shortage of water even if there was still cereal to make bread with. This led people to bake bread using wine, but the result could only be eaten warm, as it became hard as stone once it had cooled.
For these people, there are only two worries:
1) to not die of hunger and thirst
2) to not die at the hands of the enemy (meaning, the siege must not be successful)
Women and children patrolled the walls alongside the men, pelting the enemy with stones and readying cauldrons of boiling olive oil to pour over anyone who tried climbing the wall. There were accounts of women who donned their husbands' protections (reinforced leather, not armour, as these were lower class) and organised themselves to keep watch. Children would be sent to collect stones from streets and ruined houses for ammunition. Women knew full well their fate would be a little more than just death, not to mention they knew what fate awaited their children. They were up there and were hailed in the text as fierce and unshakeable as any man.
There were cases of plague at the time, so any dead person on the street could start a rumour of plague and spark panic. That meant any dead were quickly disposed of and, should plague really be suspected, they'd throw the body at the enemy camp - if they got the plague, they'd have to abandon the siege, after all.
In this case, the focus must always be two-fold. As much as people suffer hunger and thirst, they will also fear death at the hands of the enemy.
Imagine a scene where a loved one died of a disease related to the hunger. Imagine the family's pain and yet the body being chosen to be thrown over the walls into the enemy's camp. Show the loss, show the anger, show the pain and regret because the loved one will not be properly buried... but even in death, he will still fight the enemy and bring death upon them!
Unless, of course, the people blame the high class controlling the town for the siege. In that case, they may organise a rebellion or work to help the enemies inside, for as long as those rebels and their families are spared. In fact, they may even take matter in their hands, capture the noble people they believe to be at fault and deliver them to the enemy (dead or alive) in exchange for sparing the town.
The last ones to suffer with hunger. This does not mean they'd be feasting, though. They'd be controlling the food rationing to the utmost, too, but will only feel hunger at a later time.
If they live their responsibility as the town ruler / keeper, they'll suffer for the people's plight - even if only because soon the plight will be everyone's. They will make plans and remake them... because there is nothing to be done but wait and endure. Perhaps a group favours a flash attack during the night, while the others fear those men might be captured or killed, leaving the town even worse for lack of fighters. What if scenarios are checked and rechecked.
The noble person will fear for their family, too. Perhaps some, fearing the worst, are preparing ways to save their wife and children... Maybe negotiate the enemy's entry into town in exchange for the family's life? Maybe offer money? Maybe find a secret place to hide if the castle is taken over? Or perhaps both parents and children rise proudly, ready to die for their beliefs.
At any rate, fear of treason will be high. If the reason for the siege allows for it, priests and spokespeople will rally the people, painting the enemy as baby-eating demons and spurring everyone to destroy anyone who may show an inkling of desire to sell out the town into a slaughterhouse.
Amidst the noble people, there will be suspicions - Who is the least loyal? Who loves their own skin a bit too much to stand strong?
Action scenes are all very well and exciting, but in a siege, the tension is queen. Focus on it. Get inside the skin of those people.
If you want to picture hunger accurately (and are healthy), try eating nothing but a bit of stale bread at meals, alongside a sip of water, for one or two days. Feel how hunger and thirst keep you awake. It'll be easier to imagine how desperate a person would be for anything to eat in a weeks-long hunger, and yet you'll never feel how truly bad it is.
...to make the siege interesting for readers?
A siege is not interesting. It's growing levels of pain and fear until it becomes desperate hope on the edge of hopelessness. It's the will to survive, the strength to endure and the urge to fight to your last breath.
A siege is interesting to us, who are not involved in one, but a real siege is raw emotion: anger, fear, hope, despair. That is what you need to show the readers.