There aren't enough restrictions and requirements to narrow down the answer to a single build. However, I can go over your primary options and provide guidance for your decisions.
Class and subclass
For an intelligence based gish there's really only 4 options: Arcane Trickster Rogue, Battle Smith Artificer, Bladesinging Wizard, or Eldritch Knight Fighter.
An Arcane Tricksters is a great dual wielder because a bonus action attack is another opportunity for Sneak Attack and is an OK melee damage dealer because of said Sneak Attack. Spells enhance your melee damage primarily by facilitating Sneak Attack through advantage (shadow blade, greater invisibility) or an extra attack (haste). The Rogue's additional ASI helps to have high dexterity and intelligence. You get Expertise in your favorite skills.
A Battle Smith Artificer really wants to wear medium armor and control its Steel Defender with its bonus action, so it is not a good fit.
An Eldritch Knight is an OK dual wielder thanks to the Two-Weapon Fighting and a great melee damage dealer thanks to the Extra Attacks and Action Surge. Spells enhance your melee damage primarily through a small bonus (shadow blade, magic weapon) that scales with the Fighter's Extra Attacks. The Fighter's additional ASIs help to have high dexterity and intelligence.
A Bladesinger is a mediocre dual wielder and an OK melee damage dealer thanks to Extra Attack and Song of Victory. Spells enhance your melee damage through bonus damage that scales with your spell slots (shadow blade, magic weapon, Tenser's transformation). Steel wind strike is also noteworthy for being very flavorful. However, more often than not spell casting is your optimal choice rather than attacking with weapons, especially because a Bladesinger is not durable enough to be a full-time frontliner.
Multiclassing in this case will not increase your damage much, and in fact careless multiclassing can result in an overall weaker character. I'd only multiclass if I were looking for something really specific that a single class can't achieve. At the very least the following level distributions shouldn't result in a weaker character:
- Rogue 1 / Bladesinger 6 / Arcane Trickster X . Compared to a pure Arcane Trickster: slightly more melee damage and more spells, but less durable and less utility.
- Fighter 2 / Bladesinger X . Compared to a pure Bladesinger: slightly more melee damage, but slightly less spells.
- Fighter 2 / Bladesinger 6 / Arcane Trickster X . Really out there.
- Eldritch Knight 11, 12, or 13 / Arcane Trickster X . Compared to a pure Eldritch Knight: slightly more melee damage, but slightly less durable.
Races and ability scores
The build is MAD for dexterity, intelligence, and constitution. In particular, you want to start with 16 dexterity, so with that in mind, your first choice should be one of the following: air genasi; deep gnome; forest gnome; goblin; half-elf; high elf; human; mark of making human; mark of passing human; sea elf; shadar-kai; simic hybrid; stout halfling; variant human; variant tiefling.
Note that in some settings you have to be an elf or half-elf to be a Bladesinger.
Feats and ability scores
In order of importance: Warcaster; max dexterity; Resilient CON (if you are not already proficient); +2 constitution, +2 intelligence, Dual Wielder, or generically good feats.
- War Caster. Advantage on maintaining concentration is crucial because all of the options discussed above are frontliners that concentrate on a buff. The ability to perform somatic components with occupied hands is merely convenient. Casting spells as a reaction is great and works especially well with booming blade.
- Raise dexterity to 20. Most of your damage relies on dexterity, so this is crucial.
- Resilient CON. If you are not already proficient in constitution saving throws then this is also crucial for the same reasons why War Caster is crucial.
- Intelligence and/or constitution. Your secondary ability scores. Constitution is probably more important for a gish builds, but a Bladesinger might prefer intelligence.
- Dual Wielder. The extra AC is nice. The non-light weapon is so-so, especially considering that one of your weapons is often going to be a shadow blade. The ability to draw two/stow two weapons is merely convenient.
As you choose your spells keep in mind that you don't want too many concentration spells because you can only use one at a time anyway. I'll cover categories of spells and point out whether you want to know many or few such spells. I'll also point out some spells for each category, though I will focus on 4th level and below as those are the ones you cast most often.
- Cantrips. Unfortunately booming blade and green-flame blade are incompatible with dual wielding, but you still want booming blade for War Caster and for situations where you have advantage on a single attack. The rest of your cantrips should be utility.
- Damage. Enhancing your melee capabilities with spells is the raison d'être of a gish so you want at least of couple offensive buffs (shadow blade, magic weapon, haste, greater invisibility). You might also want an AoE (shatter, fireball) to be adaptable.
- Defense. Surviving in the frontlines is crucial so you want several spells that can reduce or negate damage (shield, absorb elements, mirror image, stone skin, haste, greater invisibility).
- Mobility. Getting to and from the frontline in a timely matter is important so take at least one spell that helps you do that (misty step, haste, fly, dimension door).
- Control. One or two spells that keep enemies in check (Tasha's hideous laughter, grease, hold person, slow, hypnotic pattern).
- Utility. If at this point you can still learn spells, they should be useful outside of combat.