This is a severely unoptimized build intended for beginners, that shows how you can make things work on a budget, and what optimizations you would consider as you improve things.

I picked [Broken War] because everyone gets a free Catalyzed one after completing the quest The Second Dream. Also, it's a crit-based slash weapon, which is very easy to build for and performs well in melee 3.0.

The core of this build is Blood Rush, and Berserker, two mods you can currently get for ~35p total on Warframe Market.

Even a rank 8/10 [Blood Rush] can offer crazy amounts of crit at 12x combo, which is very easy to [Reach] with the +75% speed buff you'll be getting from Berserker, and the mod used to maintain this ([Drifting Contact]) is easily farmed from Nightmare Missions.

Then it's just about increasing overall damage, and slash proc damage, because slash procs are great! Why's that? Well, slash proc damage bypasses all enemy armour and shields to apply directly to health, and it's boosted by base damage mods (like [Pressure Point]) and... you guessed it, crit!

Of course, you'll need to know how your Status Effect chance works to make those procs as common as possible. First off, the "40%" status chance this build offers is the chance of applying 1 status effect per swing, with the chance of any given effect depending on how much of that damage type your weapon has.

So while Jagged Edge doesn't directly increase slash proc damage, it does increase non-proc damage, and also increases the chance any given proc is a slash proc!

I also included [Fever Strike] and [Vicious Frost] (the latter of which you can farm from Spy Missions), which combine to add Viral damage. The Viral status effect halves the health of whatever enemy it applies to, which obviously combos great with slash proc damage!

This is just a starting point though, and there are plenty of mods you could use to improve this. The obvious ones are Primed Pressure Point for more damage, Buzz Kill for more Slash, and Malignant Force for less viral procs (you only need 1) and more slash procs! Malignant Force in-particular has a relatively easy farm vs. the other two. You could also aim for the ever-strong Condition Overload, but the drop chance on that is so low you'll likely end up spending ~50p to buy it.

Also, you might want to try changng up the elemental damage for something like Corrosive or Gas instead of Viral, especially if you're throwing Primed Fever Strike into the mix for an extra strong effect. Or eschew elements altogether and focus on pure Slash with more damage mods like Gladiator Might, and Weeping Wounds for an easy 100% status chance.

Not to mention various quality of life mods you might want, like Life Strike for easy heals at the expense of damage, or Primed Reach for hitting more enemies from further away.

These are just examples though, and it's up to you to figure out what works, preferably by checking guides by other users here who've run the numbers on all those alternatives.