One might start with the chords though with so many pieces using common chord sequences (12 bar blues, passamezzo modern, cycle of fifths, etc.) that may not provide enough information. Most of these have been used for centuries. The rhythmic structure often contributes to a song's feel.
One thing to do is listen that song and other similar songs to see what they have in common that provides the feel you like. Then try imitating parts of one with parts of others until you get something you like.
I'll give a trivial example. When trying to figure out how to write a rumba (for American or International) ballroom dancing, I examined the following songs: "Siboney" Nina Mouskouri, "Beautiful Maria of my Soul" Antonio Banderas, "Te Quiero" Mestizzo. "Harlem Nocturne" Mantovani, and "Miami Beach Rumba" Irving Fields. These cover about 70 years of composing and with different styles. I also checked other songs by the same singers (and composers) to see how what I was interested in related to other pieces.
It helps to copy out the score to the songs you like because then you see the structure from a different point of view fro just listening.