You may be wanting to ask me specifically what makes a liqueur
Wikipedia says that it's a liqueur is an alcoholic beverage made from a distilled spirit that has been flavored with fruitcreamherbsspicesflowers or nuts and bottled with added sugar or other sweetener. They are typically quite sweet; they are usually not aged for long after the ingredients are mixed, but may have resting periods during their production to allow flavors to marry.

1/4 ripe honeydew melon, peeled, seeded & cubed
1 C vodka
3/4 C white sugar, or more if you want it sweeter
2 drops green gel food coloring
1 drop yellow gel food coloring

Puree the honeydew until smooth. Measure 1 C of puree and pour into a glass jar with a tight fitting lid. You can use old wine bottles, or other. Add vodka, sugar and food coloring. Cover and shake until the sugar is dissolved. Store at room temperature for 7 days. 

Strain the melon mixture into a bowl through a paper towel lined sieve to remove any solid bits. Do this several times. Pour the melon liqueur into a bottle and seal until ready to use.

Japanese Slipper
1 part melon liqueur
1 part orange liqueur
1 part lime juice

Melon Sour
1 oz melon liqueur
1 oz sweet & sour mix
2 oz lemon/lime soda
2 maraschino cherries

Fuzzy Melon
1 oz melon liqueur
1 oz peach schnapps
splash lime juice
splash grenadine 
top with lemon/lime soda

Dirty Melon
1 1/2 oz melon liqueur
1/2 oz grenadine
4 oz pineapple juice

My Sweet Melon
1 oz melon liqueur
5 oz ginger ale
3 lime wedges