You can't, at least not by yourself.
There are points in the main quest where you have to damage an enemy to continue. Alteration has no spells that cause damage.
So, you have to pick some other way to deal damage. Since it can't be magic, it's probably a weapon in your hand.
A guide on making the best possible Skyrim character using any possible weapon is far too broad for this format.
Support Role
However, if you are okay with your follower killing everything, then it's at least possible.
I haven't tried doing this specific challenge, but I've played a game where I used Conjuration and Restoration almost exclusively. In that build, I spent most of my time standing behind my follower using Healing Hands and Courage (which is Illusion), while occasionally summoning things to help them.
You should be able to do something similar with Alteration. It's going to hamper your follower a bit if you can't use Enchanting, but maybe you can still use Smithing?
The Elder Scrolls Pages has a page detailing the Alteration school. There are useful spells there such as the various Mage Armor skills and Transmute, but I'll focus on ones you can use to help your follower:
Ash Shell and Ash Rune: These spells are available with the Dragonborn DLC. I've never used them. According to the wiki article, they paralyze creatures by encasing them in ash, but those creatures cannot be harmed while the effect is active. The rune can affect multiple targets. The effect lasts longer than Paralyze, and costs less magicka. Could potentially be used to save your follower from being downed when they are low on health.
Paralyze and Mass Paralysis: These paralyze the target for a while, preventing them from acting, while still leaving them open to attack. I haven't used these much as spells, but Paralyze is an incredibly useful effect as a weapon enchantment. Paralyze affects only one target, but Mass Paralysis affects everyone within a radius of the caster. Note that this would include your follower, making it less effective offensively, unless you're careful about the range.
The Stability perk increases the duration of these effects by 50%.
This sounds a bit tedious to me, but perhaps you'll have fun with it.
Choosing a Follower
There are many followers to choose from, and most of them are viable as long as you're within their level range.
That being said, a few are better than others. I'd recommend getting Mjoll the Lioness as soon as possible, as she should be ideal for this role. She's proficient with Heavy Armor and Two-Handed Weapons, so she can take and deal damage. She is always essential, meaning she can't die. And, to top it off, she's higher level than you.
