I love the Eldritch Knight. It is my favorite fighter class. Here is how I would build it of I was making a character for a combat focused game. If it was not combat focused and was more balanced, I would start with 13 Wisdom and 12 Constitution but otherwise the same.
Female Drow
S16 D8 C14 I8 W10 CH17 (on point buy)
Trade weapon proficiencies for tools
Level-by-level:
1. Fighting Style - Superior Technique - Menacing Attack
3. Eldritch Knight - Shield, Protection from Evil and Good, Silvery Barbs, Booming Blade, Blade Ward
4. Drow High Magic Feat, Absorb Elements
6. Magic Initiate - Warlock (Green Flame Blade, Eldritch Blast, Armor of Agathys)
7. Arcane Lock
8. Shadowblade, Fey Touched Feat (Charisma - Hex and Misty Step)
10. Alarm, Friends or Prestidigitation or Message or Minor Illusion
11. Gust of Wind
12. Shadow Touched Feat (Charisma - Cause Fear and Invisibility)
13. Counterspell, Lemund's Tiny Hut, Remove Gust of Wind
14. Fly, ASI +1 Wisdom (11) and +1 Charisma (20)
16. Resilient Feat (Wisdom), Gust of Wind
19. Fire Shield either Warcaster Feat or Mage Slayer Feat
20. Dimension Door
I love this build. It is my favorite single class fighter!
Your fighter spell selections are really limited, so for me the trick is to play a Drow, get Drow High Magic and get more spells on feats. Your offensive spells should be based on Charisma. This helps two ways first at 12th level I have 9 leveled spells I can cast for free once a day plus Detect Magic at will (and being able to do it at will is a lot better than as a ritual). Then I can use slots to cast those spells too, since they are known and the EK is a known spell caster and they are all on a charisma save DC.
My Wizard/fighter spells provide stuff that does not need Intelligence (Shield, PEG, Silvery Barbs, Absorb Elements) plus some ribbon spells.
levels 8-12 is where this really shines. War Magic is online and is not yet outclassed by extra attack. You have a good ranged attack with Eldritch Blast, which you can combine with a thrown weapon attack through Warmagic, your low level offensive spells are still potent.