The common way to create chords that fit the scale is simply to stack thirds above each scale pitch using only pitches from the scale.
In other words, let's say you're using the double harmonic scale based on C:
Let's just use those seven pitches to create triadic structures above each scale tone:
Playing around with these chords should automatically start to create the effect you're seeking.
But perhaps more importantly, you'll want to mimic Eastern progressions and melodic tropes in order to really get the sound you're looking for. A
V–I progression won't really work with this scale, since the
V triad isn't even a triad (nor is the
vii chord). Instead you'll want to use something like
♭II–I at cadences (in this case, D♭ major moving to C major), which is a really common phrase ending in this style of music.
In addition to the
vii chords not being true triads, you may want to stay away from the augmented
♭VI chord. As such, try initially limiting yourself to just
iv—C, D♭, Em, and Fm—and see if you're on the right track.