Several major factors play a big role in level 1 fights.
The base statistics of the champion
Since at level one, the tanks hasn't had a chance to build any armor, magic resist, or health yet, or the carries hasn't had a chance to build any damage items yet, everything hinges on what base stats champions are given.
When looking at the base stats, it is also important to pay attention specifically to the starting stat plus the stat gained per level. Let's look at Gangplank for example. Gangplank is generally considered a pretty squishy champion unless he builds health and armor/magic resist. He is one of those champions that people build as a glass cannon sometimes, someone who does a lot of damage but dies easily. However lets look at his base stat. He starts out with 495 + 81/level. That's a starting health of 576 plus any health that his starting item might have given him. That's quite a lot for someone who is considered a relatively squishy champion.
Now lets look at a champion that is normally considered tanky. Lets look at Udyr. Udyr's health is 427 + 99/level so he starts with 526. That is 50 less health! But Udyr has much higher health gained per level than Gangplank does. This means Gangplank has a stronger earlier game(in terms of health). You can take this thought process and apply it to the armor and magic resist and other statistics.
Skills and Passives optimized for level 1
This should be a philosophy that should apply to late game too but it is especially important early game because each of your teammates will only have on skill available to them instead of all 4. Choose a team that is balanced. You need a good balance of crowd control plus damage. A team that focuses too much on damage but has no crowd control will not be able to focus down on a single target enough to kill them. With the use of a snare or a stun, it provides a definite bull eyes sign for all your teammates to focus, usually screwing up their team formation and it will also prevent the enemy from getting away from the damage by preventing them from flashing away and making them become easier targets for skillshots.
Generally casters are a lot weaker at level 1 because they are limited to one skill and their base attack damage and attack speed are very low! This will lower the amount of damage they can deal.
When choosing a strong early game damage dealer, take a look at their base attack damage and their skills. A champion like Ganglank, has a strong early game because he has multiple sources of damage. His auto attacks can proc his passive that can not only deal additional damage but provide a slow, AND his Parrrley which has a relatively low cooldown can add a lot of additional damage.
Generally a team that is able to stop a enemy from running away or actually brings the enemy to the team will have the upper hand in team fights. Morgana's Dark Binding starts out with a 2 second snare. That is quite a bit of time to stand still. Blitz who takes his grab at level 1 will literally serve one of the enemies on a platter if one of them is pulled into your team.
Not only do we have to look at the skills a champions have but also their passives. Some champions passives will be way more useful in level 1 fights than others. For example, Volibears passive will let him regenerate 30% of his health back when he reaches a certain threshold on his health bar. Or Anivia who turns into an egg when her health bar reaches 0. Contrast that to Ryze who gets a 1 second cool down reduction for each spell he casts. But that reduction can't be utilized at all since he can only use one skill at a time. Or Ahri who gains life steal for one attack after hitting 9 targets. This is great in all for sustain but during the course of a team fight, the passive will not really play a big factor.
Sometimes in a level 1 fight, you get to an impasse where neither teams are willing to iniate and are just standing just out of range. Having a champion with a long non-target skillshot harass will generally mean an easier team fight OR allow you to push them back if you are indeed trying to invade their jungle. These skills include Nidalee's spears, Lux's Lucent Singularity(which not only damages but also provide vision for 5 seconds if she does not detonate), and Ezreal's Mystic shot.
If there is a support on your team, ask them to take Clairvoyance and to buy wards(technically, you shouldn't need to ask them if they are playing support). Vision is very important especially in early games because a champion can be focus down very fast if you can see them. If you are invading their blue, or if your blue is being invade, throw a ward onto the bush right next to where the golem spawns. Once the team fight starts, the support should throw the Clairvoyance right behind where the enemy is so provide vision of all of them. Vision in the jungle is very limited because, the bushes and the corners in the jungle will inhibit vision.
Initiation and Team Formation
Because in level 1 fights the tanks have not begun building tanky items yet, there really isn't a tank per se. Anyone who runs into the fray can easily be focused down in seconds. Because of this, it is better to stay in formation (champions with higher defensive base stats in the front, and champions with lower defensive base stats in the back), and play reactively. If one of them charges in, then focus that champion down. Generally it is smart to have your "formation" near a bush so the squishy targets can utilize that to avoid being a initial target in the fight.
Since Champions start out with 475 gold or 515 gold(if you put the two points in the ultility tree), items do not play as big of a factor. However generally buying Dorian items will give you more bang for your limited buck and give you an early game advantage.