Hadvar is one of the first characters you meet in Skyrim; his unflinching ethics, humor and admirable fighting abilities make him an attractive marriage candidate -- as long as you side with the Imperials. Unfortunately, Bethesda didn't include Hadvar as a potential marriage partner. If you're determined to marry him regardless of the limits of the game as published, install a mod that changes his character so you can lead him to the Temple of Mara where you can be joined in matrimony.
Each character in Skyrim is scripted to perform certain behaviors. For example, shopkeepers are designed to open a dialog menu during business hours that allows you to buy and sell items. Marriage partners in Skyrim are scripted to respond to you with a proposal of marriage -- if you're wearing an Amulet of Mara and have a disposition that's high enough with the characters. Hadvar wasn't scripted to be a marriage partner, so speaking to him while you're wearing the Amulet of Mara doesn't open the marriage dialogue.
Many NPCs in Skyrim share the same voice actors. Ellen Dubbin is the Many NPCs in Skyrim share the same voice actors. Ellen Dubbin is the voice for all characters scripted to respond with the female commoner voice type; characters with that voice type include Brelas, Evette San and Faida. Hadvar has a special set of dialogue and is voiced by Jonas Fisch, who recorded dialogue for Hadvar alone. Most NPCs you can marry are voiced by an actor who recorded a generic voice type or who acted as multiple characters. When you activate the marriage dialogue, that character starts speaking the recorded marriage lines. Since Hadvar wasn't intended for marriage, Fisch never recorded the marriage dialog for him.
Civil War Questline
The Civil War questline in Skyrim pits the native Stormcloaks against the Imperials of Cyrodiil, which is the faction for which Hadvar fights. If you win the civil war for the Stormcloaks, Hadvar disappears from the game. Characters who disappear or die during normal Skyrim quests are not offered as marriage candidates.
Darosh uploaded a mod called Marriageable-Follower Hadvar to Darosh uploaded a mod called Marriageable-Follower Hadvar to Skyrim Nexus; the mod changes Hadvar's scripting and makes him both a potential follower and marriage partner. Since the voice files for Hadvar still don't support marriage, Darosh gave Hadvar the Brute voice type, which includes marriage dialog. To install the file, use a mod manager or extract the mod and place the files in the Skyrim data folder. Once you've finished installing it, start the game and travel to Riverwood. You can find the modded Hadvar by the blacksmith in Riverwood.