When you are on the Character Level Up screen, the main screen where text normally goes changes into a giant graphic interface. The interface can be tricky to navigate if you can't see it, but I'll try to describe its performance the best I can:
The top-most header will tell you what level your character is.
Below that, on the left-most side, is a bar telling you how much skill points you are able to spend (something of a currency for skills). Normally this is 13 points for every level up your character has. You can choose to spend all, some, or none of the points and the rest should carry on to your next level up.
To the right of that bar, there are five rows that represent each "Core Stats" skill your character has. From top to bottom, the stats are as follows: Physique, Reflexes, Aim, Intelligence, and Willpower. Each row will have two buttons that may appear: a minus sign to subtract a point then a plus sign to add a point. By default, only the add point button is visible. After pressing the add button once, the subtract button will appear to the left of it. So it is possible to add and remove skill points to each core stat until all of your skill points are depleted. If by some way your core stats are maxed out for your level, no buttons will appear for that stat.
On the menu portion below the Level Up screen, you have a choice to Skip, Confirm, Reset, or go back. Skip will appear if you have not displaced any points and want to instead choose your class abilities instead. Confirm will appear if you have displaced at least one point and selecting this will level each stat to the amounts you've chosen and then moves to the abilities selection screen. The Reset button is listed twice for some reason, but that will reset the changes you made while displacing the points in case you messed up or want to try again. And finally, Back will take you back to the previous screen.
Confirming or Skipping your stats will take you to the Class Abilities screen. On this screen are two columns with nine rows each to which you can choose perks specific to your character class. Each class will have their own list of abilities, but you can only choose one ability for each row. So if you choose one ability from one column, the other ability of the opposite column will be disabled the next time you choose another perk. Additionally, each column's row will unlock once per level starting from the top going downward. So if you are at level 2, only the top row is available for both columns. If you are level 5, the top four rows will be available depending on whether or not you have already selected an ability for any of the available rows. Clicking one ability will select it and allow you to read the ability's description. Clicking the same ability again will deselect that ability.
Similar to the the previous screen, you will have a perk point currency in where your character will only gain 1 perk point per level up. You can save up these points if you choose not to spend them on any class abilities at the time of leveling. To continue, just hit the Confirm button, and any ability you have selected will be placed into your character's perks list. After this, your character will also automatically gain another passive class-specific perk for leveling up.
This will complete the skill and perk leveling process. I'm not sure how accessible these interfaces are, but I hope what I explained helps!