The screen on my laptop I checked in has stopped working. Could I claim something? [closed]

First I would say you should contact your insurer, although it is unlikely it will cover you checking in a laptop.

Your alternative recourse is to threaten to take BA to a small claims court. I will give a brief explanation of why you are entitled to do this (despite the contract purporting otherwise), but I think you will not succeed because the laptop is likely to be seen as inherently fragile.

However, if you send BA a letter before action, they may decide to negotiate with you instead of risk the court case going badly and risking future actions.

In principle the carrier on an international journey is strictly liable for all proven damages while the baggage is under its control, up to approximately 1400 USD. This follows from the Montreal Convention (done in 1999).

The Convention has had force in US Law since 2003. https://www.congress.gov/treaty-document/106th-congress/45/resolution-text

Liability for carriage of baggage is given in Article 17 (2)

2. The carrier is liable for damage sustained in case of destruction or loss of, or of damage to, checked baggage upon condition only that the event which caused the destruction, loss or damage took place on board the aircraft or during any period within which the checked baggage was in the charge of the carrier. However, the carrier is not liable if and to the extent that the damage resulted from the inherent defect, quality or vice of the baggage. In the case of unchecked baggage, including personal items, the carrier is liable if the damage resulted from its fault or that of its servants or agents.

(Emphasis mine—unfortunately I think the "inherent quality" test is not in your favour, but I might be wrong.)

The carrier's liability is limited to approximately 1400 USD by Article 22 (2)

2. In the carriage of baggage, the liability of the carrier in the case of destruction, loss, damage or delay is limited to 1,000 Special Drawing Rights for each passenger unless the passenger has made, at the time when the checked baggage was handed over to the carrier, a special declaration of interest in delivery at destination and has paid a supplementary sum if the case so requires.

[A Special Drawing Right is fixed by a basket of currencies, but it works out at 1400 USD: https://www.google.fr/search?q=1000+XDR+en+USD .]

Finally Article 26 invalidates the BA contract of carriage so far as it purports to restrict its liability.

Article 26 - Invalidity of contractual provisions

Any provision tending to relieve the carrier of liability or to fix a lower limit than that which is laid down in this Convention shall be null and void, but the nullity of any such provision does not involve the nullity of the whole contract, which shall remain subject to the provisions of this Convention.