What Romantic Chemistry Looks Like

Relaxation: Body language and facial expression will be relaxed, open, and "warm" when they're around each other. Their guards will be let down.

Eye contact: The pair can easily and comfortably make eye contact with each other. (Assuming one or both doesn't have an aversion to eye contact.) They easily and readily face each other when speaking to each other.
Watch out! Do not confuse hostile eye contact with romantic eye contact! If you're watching characters interact with each other in a visual medium, pay attention to the rest of their body language - is it warm and open (EG, arms down, relaxed facial expressions), or cold and prickly (EG, arms folded, tense facial expressions)? Also, pay attention to what the characters say - do they say nice, positive things to each other, or do they just insult and snipe at each other? If the characters are cold, prickly, or rude, you're probably looking at hostile eye contact.

Getting physically closer: They lean in closer to each other when they're talking, or are otherwise comfortable being in each other's "personal bubbles."
Watch out! Getting in each other's spaces can sometimes be a hostile move. Are they happy and relaxed when this is going on, or are they tense and at odds with each other? If it's the latter, it's probably not romantic chemistry.

Positive speech: They praise or admire each other's accomplishments and give genuine encouragement and compliments.

Happiness: EG, they're talking together, playing a game together, and are clearly happier for having the other nearby. They may crack jokes and laugh over them together, and they may find each other's laughter infectious. When they look at each other, their expressions will show happiness and warmth.
Watch out! If one or both of the characters are only happy when the other is present, this is a sign of an unhealthy relationship.

Compromising: They take each other's feelings and opinions into consideration and make accommodations for each other.
Watch out! Does this genuinely go both ways, or is it always one partner who does the compromising? If so, you have an unequal and unhealthy relationship.

Favors: Be it through gifts, doing chores for the other, showing extra consideration or thoughtfulness to the other, or being more forgiving of faults in each other than they would be in others.
Watch out! It should never be shown or implied that one character owes the other a relationship because of favors done. It should be used to demonstrate that attraction simply exists, or to show appreciation of an existing relationship.

Empathy/sympathy: When one of them hurts, the other feels badly and will willingly offer support or help. When one feels happy, the other shares in that joy.
Watch out! Again, does this go both ways? If not, you have a very one-sided relationship.

Awareness: They take notice of each other, even in relatively crowded places, because their subconscious minds are on the lookout for each other.
Watch out! This should not happen if they don't already know each other and have other forms of chemistry already. If the characters are still strangers, they shouldn't notice each other more than they'd notice any other random stranger.