Find the root note of the chord you're going to construct. The title of the chord is its root note. For example, the root note of A minor is A, the root note of B major is B, and so forth. Start by using middle C if it's easier for you.
Using C major because it's the simplest, count three whole notes (white keys) away, including the note you're on, to arrive at E. C is one, D is two, and E is three. Having your thumb on C and your middle finger on E, find the fifth.
Count five notes from C, using C as one, to arrive at G for your fifth. You should now have your thumb on C, your middle finger on E, and your pinky on G. This is the C major chord.
Know your keys to build chord structures. These same basic steps can apply for building any major chord, as long as you know what key you're in. For instance, A major has three sharps: C sharp, D sharp, and G sharp.
Taking these same basic steps, you can experiment and find chords based on their distance from the root note. Using C major again as an example, and instead of building the triad, we could start with the C root note, add a third, and instead of adding the fifth, we could add the seventh, B. This give us C7.
When you finish learning how to build chords in major keys using the triad for your basic blueprint, you can build different chords in other scales like the minor scales, and even in modes. Apply the same basic steps as above but use the appropriate key and scale to find your notes. In G minor for instance, the minor third and fifth would have to be found to build the chord. Just look on your circle of fifths to find the sharps and flats for G minor and you'll see that the minor third is A sharp instead of B as in the major chord. It may only be half a step away, but the feelings the chords create are worlds apart.