It is possible to sue for personal injury by yourself, without a lawyer. While this article is by no means legal advice, it does have a lot of useful information to get you started.
In California, the definition of personal injury is covered by the California Civil Code, section 1714.
"Everyone is responsible, not only for the result of his or her willful acts, but also for an injury occasioned to another by his or her want of ordinary care or skill in the management of his or her property or person…"
There is an exclusion about injuries which you cause yourself ("willfully or by want of ordinary care"). There is also an exclusion about injuries and alcohol (details in the link above).
If your situation still fits this description, then most likely, you have a personal injury claim.
In legalese, "standing" is their way of describing whether or not you are qualified. So, what makes you qualified to file a personal injury suit?
Read more details about this point in this California court post.
When did the accident happen?
In general, people need to file their claims within two years of the occurrence [Source 3], even less if the claim is against a government body such as a city or state agency in which case it is 6 months [Source 4].
ARTICLE 1. General [910 - 913.2] ( Article 1 added by Stats. 1963, Ch. 1715. )
911.2. (a) A claim relating to a cause of action for death or for injury to person or to personal property or growing crops shall be presented as provided in Article 2 (commencing with Section 915) not later than six months after the accrual of the cause of action. A claim relating to any other cause of action shall be presented as provided in Article 2 (commencing with Section 915) not later than one year after the accrual of the cause of action. - Source: California Legislative Information (Source 4)
You now need to figure out who you should be suing. This is not always as straightforward as it sounds. Let's look at two examples.
A car accident
Most of us would think that we should sue the driver of the car that hit us. We would often be correct, but what if the driver was not the owner of the car? In that case, we should also file a claim against the car owner since the insurance will be in that person's name.
A fall in a restaurant
Is this a chain eatery? If so, then we need to file our claim against the chain. If it is just an independent restaurant on its own, we will need to file against the owner(s) and not an employee. Keep in mind: the manager is not always the owner.
As you can see, your claim needs to be against the correct party in order to be valid. If you don't get this part right, then your claim may be dismissed because the party you sued is not at fault.
First, you need details and documentation about the accident or situation that caused the personal injury. The more objective proof you have, the better. In other words, not just your say so, but facts and figures from relevant, impartial parties such as doctors, garage mechanics, or structural engineers. Include visual media (photos, videos).
Next, you need all the necessary details about the party you're suing. There are tips on how to find these out in this California court post.
Lastly, it is wise to follow the correct filing procedures. Even the best claims will fail if papers are not filed according to court rules. This California court post has more details.