You need high HP for your skills and this is a tank build, so no -hp or -def IV. Falchion because it's free, it heals and hurts- especially dragons. Reposition for mobility or reciprocal aid to heal others or yourself. Bonfire/Ignis capitalizes off Chrom's huge Def stat, Aether is free and gives you better arena matchups and heals you. EARTH BOOST is the cornerstone of the build, gives you a huge def boost and synergizes with your free heal/hurt sword. Quick riposte lets you use your big free sword twice on enemy phase. Hp Seal because you want to have more HP than your opponent for earth boost.
OVerview: This build was designed keeping in mind that most of us aren't about to go burning 5* heroes to top off our arena team- ESPECIALLY 5* exclusive heroes like Ike for sword breaker 3. This build gets the most out of falchion and doesn't rely on any skills you can't inherit from a 4* hero. This Chrom is an absolute wall and hits like a truck in return. He's a great tank, a pretty decent OHKO nuke and depending on the build can be an off-healer or bodyguard. I ended up having to use Chrom in my arena team because I've somehow managed to draw 5 5* Chroms throughout my F2P FEH career. So, I had to make him viable in arena to take advantage of the better matchups his + levels helped generate. After some trial and error I landed on a build that has made him a consistent contributor to my arena wins, often the MVP. As he stands he bests ALL axe and sword matchups (one or two flukes not-withstanding, usually involving a brave sword and unfortunate timing) and many/most spear match-ups. He can always bait mulagir Lyn out of position and it's about a 50/50 with Brave Bow lyn depending on specials, buffs etc. just remember to do some math. Also He has a fighting chance against dragons and mages in a way that most Chroms might not.
IVs: This Chrom functions off a high Defense and HP stat, this build is NOT viable with -def or -HP but can work with -atk, although it's not ideal. I prefer to lean into his abominable resistance stat and just keep in mind that he can NOT take a hit from a mage or dragon, especially if he's being doubled- his high HP can often eat a single red or green magic attack but almost NEVER 2. His speed is bad, but with neutral speed and some buffs he often avoids getting doubled. With neutral RES you have better odds at taking a hit from a dragon, especially Fey, but god help you if a Nowi takes a shot at you.
Weapon: Falchion has the highest MT of any sword in the game, is super effective against dragons and heals 10 hp every third turn, which is crucial for keeping Chrom in Earth Boost and Quick Riposte range. if you have a brave blade you can spend on Chrom, there are plenty of great brave blade builds that put it to good use: but like I said, this build is about being affordable and not blowing a 5* hero to inherit a weapon. This build is all about eating hits and getting extra hits in on enemy phase. Two strikes from Falchion fucking huuuuurts. Plus, if you can get the drop on a mage and especially a dragon, OHKOs are pretty common. Chrom is the kind of unit you can over extend on offense to take out a glass cannon, and just leave to let the enemy team break themselves off your face. I have a brave sword to give Chrom right now, but I prefer this build over what a brave build would look like.
Assist: His assist is reposition on my team when I have no other troop repo skills available. My common tactic is to move forward with a glass cannon, blow someone up, have Chrom come up from behind to repo the cannon out of the way and run defense (and usually kill anyone that comes at him). This is body guard mode- keep my nino safe and land her next to my eirika. When I have Camus with reposition on my squad, the other option is reciprocal aid: enabling him to off-heal if necessary, or steal health off an ally if he's brought real low and needs to quickly bop back up to Earth boost or QR range.
Special: Bonfire and ignis procs are enormous when you're also benefiting from earth boost- easily breaching 40 def without even needing buffs from any allys. I prefer bonfire because of the lower cooldown and it makes surviving a standard enemy trade really unlikely for your opponent: enemy attacks (2), Chrom attacks (1), enemy attacks (*) Chrom attacks with bonfire dealing ~20 extra damage. No one survives that. Ignis works too, but a little differently: Enemy attacks (3), Chrom attacks (2), enemy attacks (1), Chrom attacks(*) and you'll start the next turn with an ignis in the chamber to drop 32 extra damage on whichever unsuspecting fool happens to be in range. But things don't often work out this neatly anyway, so it's really up to personal preference and I think bonfire does enough damage. That being said: I'm currently using Aether on my Arena team- mostly because as a third tier skill it again helps bring in higher rated arena matchups, so I take what little advantages I can find, where I find them. But even so, despite it's stupid high cooldown, the Sol aspect of Aether does help to keep Chrom high in health, which as I have already stressed is important for this build.
A skill: Earth boost is the sleeper hit here, you can inherit this bad boy off any 4* Bowie, so it's not hard to get your hands on. This is why a +HP or neutral HP Chrom is important. It's not enough to have a high percent of hp for your B skill, you also need to have a higher number HP for your A skill, which is much safer and easier to proc than defiant denfense which gives the same huge bonus. With neutral HP and an HP seal, it's really easy to get your HP to ~50 which is higher than you're ever likely to see on an opponent. A full health Chrom with earthboost is almost unkillable with physical weapons. This skill also synergizes really well with falchion's passive heals. If you're brawling it out with an enemy and they happen to get you lower in health than they are, voila, +10 healed and you're back in earthboost range. Can't speak more highly about Earth Boost.
B skill: Quick Riposte 2 is my preferred B skill. 4* subakis are not hard to find. 80% feels high, but it's pretty easy to keep your HP above 40, especially since most opponents striking you will in melee range will barely make a dent in your HP anyway, so why not let them deal another 2 damage and give you an extra special charge? Why keep them from attacking you if they're hardly doing any damage anyway? However, Sword breaker is helpful against that rare 40 +10, brave sword and luna ogma or Hana that might pop out of nowhere in level 19 and 20 arena. But typically there's virtually no one in the meta that requires you to have sword breaker, but it might be considered the safer option in that it prevents your faster brave sword weilders from quadding you and proc'ing their special.
C skill: C skill is flexible, and should reflect what you need in your team if you're looking for a buff to give him. But if you do like to drop your chrom in the middle of the enemy team and let them kill themselves off you, threaten attack or speed would help weather that storm. Personally I like having spur defense to help keep your buds safe.
Sacred Seal: Lastly HP seal is crucial to getting a high max health to facilitate having higher HP than your opponent to enable earth boost.