What is needed for a soldier to advance his/her psionic abilities?
Just use their abilities in battle as much as possible.
I haven't seen any math suggesting how the progression works. I have a level 3 psychic who has used uprwards of 6-7 abilities in total. she has secured 3 psi kills iirc. Last mission she used an ability once and only damaged and enemy but that leveled her to 3.
With regard to gaining psionic powers in the first place, soldiers with higher willpower are more likely to become psionic when tested, for full details see What determines if a soldier is gifted with psionic powers?
Once a soldier has gained psionic powers, there are two important factors: the soldier's willpower (which determines how likely the soldier is to succeed when using a psionic skill), and psi promotions, which unlock additional psionic powers.
Each point of willpower increases your chance of succeeding when using a Psi attack by one percent. Willpower increases each time your soldier advances in rank, gaining a normal (non-psionic) skill. The gains are higher if Iron Will has been purchased from the officer training school, so soldiers trained before Iron Will is unlocked will never be as strong as soldiers trained afterward.
Taking a critical hit lowers willpower permanently, so soldiers that spend time in the line of fire usually have their psionic abilities decline over time once they stop earning promotions; however, an Assault can choose the Resilience promotion when they reach Colonel, giving them immunity to critical hits, and preventing their willpower from degrading.
Psionic promotions are earned by using psionic skills in battle. The number of missions necessary to gain a psi promotion will depend on how aggressively you use psi powers, and on how large each mission is, but in my experience, on normal difficulty, it's possible to get the first psi promotion in just one mission if you use Mindfray every time the cool-down expires, and the second and final psi promotion can be gained in just two additional missions, if both psi skills are used regularly.
Just don't use your Psi skills so regularly that you get somebody killed by using Psi Inspiration when you should have used Double Tap, Rapid Fire, or Bullet Swarm.
I should note that I usually take Psi Inspiration, which is easy to use regularly between battles to keep willpower high, boosting Psi attack success rates, and making panic less likely. It may or may not take longer to level when using Psi Panic, which has to be used tactically (i.e., you need an alien to use it on, and it may be risky to use on aliens that have a good shot on your team).