I’ve played as a mage, warrior, etc. Now I really want to play as a race that’s good at smithing, but I don't know which race I should choose.
tl;dr: Any race you want.
As you can read on the Skyrim Wiki article about races, most races start with a smithing skill of 15 while the following races start with 20:
- Orsimer (Orc)
Of all those races, Orc also comes with 20 instead of 15 in Enchanting, which synergizes very well with blacksmithing.
But the racial bonuses to starting skills are not really relevant in general. Those 5 skill levels are trivially easy to achieve on your own as soon as you are in the game. What's far more important are racial powers and racial passive effects, because those can not be substituted that easily by what you can obtain throughout the game. But there aren't any which have any obvious synergy with blacksmithing.
So bottom line, your race choice is mostly an aesthetic one.
But you are not going to achieve much in Skyrim just with blacksmithing alone. There are very few quests you can solve just with crafting or just by spending the money you make through crafting. Even just obtaining the crafting materials for blacksmithing will require various combat skills. So you also have to consider what to spec into in addition to blacksmithing. The most obvious playstyle choice is of course a melee fighter, because it synergizes most with blacksmithing. But which build in particular (light or heavy armor? Two-handed, shield or dual?) is a matter of personal preference.