No, your Dexterity bonus can't be used for grapple checks instead of Strength.
You don't get the swapping benefit for everything you do unarmed — being unarmed is just the first condition of the ability. You get the effect, when unarmed, for "unarmed strikes and monk weapons."
Starting a grapple isn't striking (aka hitting to deal damage), it's grabbing them to hold on.
The basic principle is that not all melee attacks in the game involve actually hitting someone in the "striking" sense. There are lots of melee attack rolls called for in the rules that are there to resolve something other than a strike.
Now, if you want to throw someone in a martial-arts style after a successful grapple, or grab someone to throw them instead of to impose the Grappled Condition, that's an entirely different ball of wax because grappling doesn't offer that option, but there is another rule that does.
For that you actually want to look at the improvisation rules (PHB, p. 193), which invoke the DM's Contest rules (DMG, p. 238), in order to resolve a throw. If your throw technique involves speed and agility to maximise leverage to use the opponent's own mass and strength against them instead of using your own Strength, that would reasonably qualify as a Dexterity Check for your half of the Contest.
TL;DR: Yes, he can attack if he has enough Movement to make the climb and does not have to use two interactions to both stow and draw a weapon.
Walls can be covered under "special types of movement." (PHB, p. 182). The rules-as-written allows that attack if he has enough movement left to climb that distance, since you can generally move and attack in the same turn.
Climbing, Swimming, and Crawling
While climbing or swimming, each foot of movement costs 1 extra foot (2 extra feet in difficult terrain), unless a creature has a climbing or swimming speed. At the DM’s option, climbing a slippery vertical surface or one with few handholds requires a successful Strength (Athletics) check. (Basic Rules, p. 64 / PHB. p. 182)
If he had at least 20' of movement left, it fits RAW that he'd complete the move up the 10' wall and then attack, providing the character succeeded in the Strength (Athletics) Check you required. Since you have already determined that the wall isn't that hard to climb, there seems no reason not to allow the attack. If you had ruled it difficult terrain, he'd have needed 30' of movement to make the climb and then attack.
Other Activity on Your Turn:
[…] You can interact with one object or feature of the environment during either your move or your action. (PHB, p. 190, Basic Rules, p. 70)
What you may wish to rule on is whether or not he must
- stow the weapon (interaction)
- climb the wall (movement)
and then draw the weapon (interaction)
If he needs two interactions (only one interaction is free) that would most often preclude the attack. (In some cases, a bonus action could still allow it.)
You state that this wall is not hard to climb. You may or may not allow him to climb this wall with a weapon in hand. You could increase the DC of the climb if he attempts it with weapon in hand. As that wasn't specified in the question, I'm can't suggest a ruling either way.
Your Turn
On your turn, you can move a distance up to your speed and take one action. You decide whether to move first or take your action first. Your speed—sometimes called your walking speed—is noted on your character sheet. (PHB, p. 189; From Basic Rules, p. 69)
Movement and Position
You can use as much or as little of your speed as you like on your turn, following the rules here. Your movement can include jumping, climbing, and swimming. These different modes of movement can be combined with walking, or they can constitute your entire move. However you’re moving, you deduct the distance of each part of your move from your speed until it is used up or until you are done moving. (PHB, p. 190; Basic Rules, p. 70)
If the character is a Rogue with Second-Story Work (Thief Archetype) then he'd not need all 20 feet of move, just the 10 feet.
Second-Story Work
When you choose this archetype at 3rd level, you gain the ability to climb faster than normal; climbing no longer costs you extra movement. (PHB p. 97;
Basic Rules, p. 28):
Best Answer
So I think there's two problems that have to be answered in your question: whether or not climbing can invoke an ability check, and whether or not climbing should require your action.
Whether or not a particular character behavior requires an ability check is dependent on the DM, unless the rules have reified that behavior as requiring an ability check. So it's fully within the DM's purview to say "climbing this sheer cliff requires a Strength check, you can add your athletics proficiency", or to elide that check for any reason at their discretion.
However, climbing a cliff is governed within the general rules on movement:
As you can see from the quoted rules here, there's already a cost baked into the act of climbing, whether or not it includes making an ability check to successfully do it. Although the DM is technically allowed to impose Action Economy costs on top of this (as they are with any other behavior not explicitly reified as such in the book), they really shouldn't have in this case. Climbing is a type of movement, and should be treated as such.
My best guess, without personally knowing your DM, is that they may have been thinking of rules from other editions of D&D, or possibly other d20-based games (like Pathfinder) where such checks may have required the use of the character's action. My guess is based on the colloquial use of the term "skill check", which don't exist in 5e anymore―5e has "ability checks", which may or may not add your skill proficiencies to the roll depending on the type of check or the DM's discrepancy. Before your next session, I would talk to your DM about revising this case (and other cases that involve special movement speeds).
One other thing I'm going to add, because I saw it mentioned in comments below and figure it's good advice that should receive visibility: if the DM is going to associate an Action Economy cost to behavior that isn't reified as such in the books, they should convey that to the player before the player makes the decision to do it. I can understand unusual circumstances where an action to climb over a wall might be appropriate (perhaps the DM is inferring the player is taking the Dash action to get some extra distance?) but if that's the case, the DM should be stating that outright so that the player can make choices about how they want to spend their turn.