There are two things to consider here: (1) the player's demand; (2) the character's demand. If you wanted to multiclass to a cleric, you could easily say that your character is going through an existential crisis and then head for the nearest temple for guidance. But in the case of a class like warlock, the answer might not be so easy if your character has never met someone who is a member of that class.
Since the warlock-patron bond is highly setting dependent, it is best talking to your own DM for specifics. Here I will try to offer a few examples from the Forgotten Realms setting, as it is the "default" setting to which a large number of the 5th edition sourcebooks regularly refer.
If it is your demand to play a warlock, but your character has not encountered any warlocks or possible patrons before, you need to declare to your DM that you want to multiclass and she might come up with an event that will naturally push the "destiny" of your character towards becoming a warlock. For example, you or one of your party members might accidentally summon a being from the outer planes. This is the story of Farideh from the "Brimstone Angels" series; she is a tiefling who met her patron when her twin sister's spell to summon an imp went wrong. A short summary, as well as Farideh's 5th edition stats are available here.
If it is your character's demand to become a warlock, because he has heard about or encountered warlocks before, then you should tell your DM what he (your character) is actively doing for this. He might try to find ancient tomes in libraries, or seek hallowed or desecrated grounds for praying, or get in touch with known warlocks, or groups that are likely to have warlock members or associations which might be in the service outerworldly patrons. Examples could be the Host Tower of the Arcane, the Cult of the Dragon, etc. Maccath the Crimson from "The Rise of Tiamat" adventure module appears to be such a warlock according to the FR wikia.
FR wikia offers a couple more of warlock backstories under the Warlocks (5e) article.