Hunger
Throughout the film, we see The Tramp, McKay, and Larsen having to deal with the extreme conditions in Alaska while they are in search of gold. The literal issue of hunger as the men run out of food is seen clearly when The Tramp eats a candlestick and Larsen is upset by his not sharing. And we later see that hunger has lead The Tramp and McKay to eat one of The Tramp's shoes in order to stay alive.
In a figurative sense, the hunger of the American people to seek and find a better life through gold mining in Alaska is also driving throughout the film. We see McKay primarily as the one seeking his fortune, but we also have images of people climbing the mountains seeking a way to have a better life. Also, we see how towns are built overnight because of the influx of people to a community during this gold rush period thus a new way of life is created nearly instantly.
Trust
One of the primary themes in the film is that of trust. We see it initially between The Tramp, McKay and Larsen as they share a cabin in the wilderness and snow. They trust each other to find food for one another, and that trust is broken when Larsen doesn't return with the supplies he steals and keeps them for himself. We then see the trust that The Tramp has in Georgia and her friends after they come to the cabin he is watching and make plans to come for New Years Eve. Again, the trust is broken.
By the end of the film we see that trust is regained as McKay entrusts millions of dollars to The Tramp for getting him back to his Mountain of Gold. Georgia and The Tramp come together on the ship kissing in the final shot, which shows the trust that was broken between the two has been restored.
A Better Life
The reason people are heading to Alaska is to gain a better life for themselves and their families by striking it rich with gold in the mountains. Chaplin shoots this theme brilliantly on top of a peak, where we see a line of people climbing to the top of a mountain with a string of many more following them below. The opportunity to go from no class to high class was realistic if someone struck gold. They would be able to gain the respect of society, provide a better way of living for their family and earn the ability to pull the next generation out of poverty and into opportunities to do even better than they had done.
