tl;dr: To live an altruistic life understood as unconditional self-sacrifice would very soon make you unable to care for anyone at all. Instead, I define altruism as the imperative to increase the well-being of all sentient beings. In this way, altruism implies that you should put your own needs first. I strongly believe that you can bring more than just one life to fulfillment.
Epistemic state: I occasionally thought about this over the last years. Even though it contains strong beliefs, this is still a momentary snapshot that might change at some later point.
When I was younger, I thought altruism does not exist. People often seemed to encourage helping others without being rewarded for it, but it just didn't make sense in my head. In my mind, the only reason someone ever would care for someone else is because they are rewarded for it in some form. Either by directly getting something back from others by (generalized) reciprocity, or by rewarding themselves, e.g., for being loving or powerful. But there is nothing flowing into a decision that calls for helping others without being rewarded.
Furthermore, I'd even argue that one can't value anything outside of oneself. We hold beliefs about the world in our minds. These internal representations are the basis of our experience. Any new perception is immediately pre-processed and filtered before something reaches our awareness. That means that everything we can ever optimize for are internal states of our brain, never the actual outside world. Since others are parts of the outside world, we can never truly care about them, but only about our internal representation of them. Give me cogent evidence that everyone is well and I will believe it, no matter what the actual state in the world is!
I even think that helping others without being rewarded is an unreasonable demand in the first place. Firstly, almost all available actions can be somehow interpreted to have some selfish gain, such that it is really hard to find any action that actually follows this goal. Secondly, think about what would happen if you actually followed it all the way through, in that you'd only ever allow actions that actually don't reward yourself. You'd die within a few minutes because of insufficient supply of oxygen! Even if you only count the "controllable" actions, you'd face similar issues and probably die within a few days or weeks. This seems intuitively like a really bad outcome. In case this is what is meant by altruism, then nobody can possibly be completely altruistic. Unconditional self-sacrifice is a non-sense concept.
I see the actual source of the imperative of "helping others without being rewarded" in social interaction. Firstly, actions that involve helpfulness can be used to signal cooperative behavior. Since cooperation is often advantageous, such behavior is intuitively desirable. But what you actually want is to take advantage of the reciprocal cooperative behavior of others to you. Secondly, people already in power can demand such behavior to ensure their power stays unchallenged. Helping others without sufficient rewards is a net cost for the one doing it, straining their resources, ultimately decreasing their power.
Let me try with a saner definition of altruism: Remove the "without being rewarded" part, i.e., altruism means helping others. Now you're allowed to take actions that also involve personal gain. At first it might seem that when following this one through you'd still neglect yourself, similarly to above. You'd do whatever helps the other most in any given moment and only take personal gain by accident. But this is myopic. You'll be able to help people much more in the future if you survive longer. You'll be able to donate much more to poor people if you are a billionaire. Whatever your high-level goal is, you are probably better able to achieve it if you have more power. Altruism implies personal gain, because power is instrumental to help others. In a game theoretic sense, altruism does not mean always cooperating and never defecting.
Okay, but how is altruism then different from true selfishness? It is in what you value. Altruists are more likely to choose actions that are valuable for others. Imagine you are on a walk and suddenly see some significant amount of money lying on the floor. Now, you have the choice to take the money with you or leave it there for the original owner to be found. As a (utilitarian) selfish person, you might weigh the things you could additionally do with the money for yourself against the expected risk of being caught for theft, and might decide to take the money. As a (utilitarian) altruist, you might also include the owner's suffering in your calculation, and might decide to leave the money (or take it to a police station). More generally, a pure altruist would use their power (like their working time or wealth), to support people irrespective of whether it would make themselves happier, while purely selfish people would rather use their power to their own enjoyment.
Let's take a step back and understand what I mean by helping others. Help refers to increasing well-being, by which I mean to fulfill needs of the person in question. Say, an elderly person wants to cross a street to reach the supermarket, but can't reliably perceive approaching cars. Then it might be in your power to increase the likelihood that the person reaches the supermarket and can buy food. Others can refer to anyone you consider as part of your moral circle, which is the beings whose well-being is important to you. I find it appropriate to choose all sentient beings for this. In summary, altruism is the imperative to increase the well-being of all sentient beings.
To be clear, even though I do sympathize with effective altruism, I definitely also think of myself as partially selfish on a fundamental level. And to the extent I care about others, I generally tend to care more about humans than about animals, and more about animals than plants. Nevertheless, I strongly believe that there is inherent value in the well-being of other sentient beings.
There is the common trope that self-care is for everyone. While I think this is true, I also think most of my readers can do better than just fulfilling the needs of one human life. We live in a world of abundance, where we can attain enough power to help ourselves and others. Even as an altruist, it is not only allowed, but necessary, to put your own needs first. But as far as you can choose in the remaining options, do whatever helps others most.