We know how to attach $p$-adic $L$-function to elliptic curves over $\mathbb Q$ only because we know how to attach them to cuspidal modular eigenforms, and we know
by Breuil-Conrad-Diamond-Taylor that every elliptic curves over $\mathbb Q$ are themselves attached to a cuspidal modular eigenform of weight $2$. Historically, the construction of $p$-adic $L$-functions attached to modular forms is the work of Mazur-Swinnerton-Dier, Manin, Amice-Velu, Shokurov. The article of Mazur-Tate-Teitelbaum quoted in comment contains a synthesis of all that.
But I think the best way to learn the theory now is to learn the version of the construction (not fundamentally different than the previously mentioned, but much simpler and clearer) given by Pollack and Stevens. For this, see the two papers by Pollack-Stevens:
Overconvergent modular symbols and p-adic L-functions, Ann. Sci. Ec. Norm. Sup. (4) 44 (2011), no. 1, 1–42, and Critical slope $p$-adic $L$-functions, J. Lond. Math. Soc. (2) 87 (2013), no. 2, 428--452.
I have also written a book, an expanded version of a course I gave at Brandeis, on the subject of p-adic L-functions of modular forms (and families thereof). It exposes the Pollack-Stevens method, as well as their generalization and mine for the so-called "critical" case, that was missing at the time of Mazur-Tate-Teitelbaum, and many other things.
The book is called The Eigenbook. Eigenvarieties, families of Galois representations, p-adic L-functions and it will be published soon by Springer-Birkhauser, but you can also find a non-final version on my webpage.