Ooo, What's this potion do?

[Biffoniacus_Furiou]

Flavor is free. Since you can use your alchemist's supplies as a spellcasting focus, you can absolutely flavor it as downing a flask as part of the spellcasting.

Regarding not falling behind in melee, the dueling style isn't going to be enough IMO. You don't get extra attack, so casting Green-Flame Blade with Alchemical Savant to add your Int bonus to the damage is the best way to accomplish this. The Tools Required portion of your spellcasting feature says, "You must have a spellcasting focus—specifically thieves’ tools or some kind of artisan’s tool—in hand when you cast any spell with this Spellcasting feature (meaning the spell has an ‘M’ component when you cast it)." Thus your spellcasting feature adds an additional material/focus requirement to every one of your spells. As long as your alchemist's tools are used for that you'll always benefit from Alchemical Savant on every spell that deals the right type of damage. Maybe you shake up a flask and pour an unstable burning concoction on your sword before striking with it, some of the burning mixture is even flung off onto a second target.

If you'll be holding a weapon and your alchemist's supplies, that would exclude using a shield. However, a Cleric is specifically able to use their shield as a spellcasting focus and cast spells while holding it if their other hand is full if their holy symbol is emblazoned on their shield. You should be able to carry a rack of vials that doubles as a shield to gain the same benefit.

You can absolutely optimize your background choice. If you can use a Strixhaven background to gain a feat that's even better than Magic Initiate, definitely do that. If you can use a Ravnica background that adds more spells to the ones you can access, go for it. The PHB specifically says you should customize your background, and then presents the printed backgrounds at examples of what's possible. Always swap out the background proficiencies to something more appropriate/optimal. Swap the background feature for one more fitting. You could pick a Ravnica background, use that background's starting equipment and expanded spells, swap the feature for a Strixhaven background's feature which grants the Strixhaven Initiate feat, and swap the skill and tool/language proficiencies for better ones (always get Arcana for making scrolls, always get Perception and Stealth proficiency as those come up most often). For tool proficiencies, be sure to pick useful tools and avoid redundancies with the rest of your party.

For race optimization, go with a Mountain Dwarf and replace all four weapons and both armor proficiencies for six more tool proficiencies per Tasha's. Or keep either the battleax or warhammer proficiency and use that instead of a longsword. Put your two +2's in whatever stats you want.

For your ASIs/feats, get War Caster for sure. Another Green-Flame Blade instead of a normal opportunity attack, plus the concentration bonus, is well worth it. Spend your other ASIs to max out your Int and maybe even Str and/or Con, maybe get Resilient: Wis or Lucky or Alert besides those.